Modification of the Kedem-Katchalsky equations

The presented modification of the transport equations of Kedem-Katchalsky resulted in the introduction of (omega s/omega) and omega/(omega-Lp sigma[(1-sigma)C1-(1-sigma s)C2]) factors into the Kedem-Katchalsky equations. The above factors determine the influence of boundary layers on transport across the membrane. The modified Kedem-Katchalsky equations were verified for synthetic membranes and it was shown that the value of the (omega s/omega) factor depended on the type of membrane and the membrane configuration system. This modification facilitated a wider range of application of the Kedem-Katchalsky equations to systems in which the solutions were stirred or unstirred.     

Measurement of membrane potential and estimation of effective fixed-charge density in membranes.

Electrical potentials Em arising across cross-linked phenolsulfonate membrane separating NaCl solutions of molality M1 and M2 have been measured at 25 degrees C. These values of Em have been used in the Nernst equation to calculate values for the apparent transport number ti(app) for the counterion or the co-ion in the membrane. Values of ti(app) together with the limiting value for the cation transport number in the aqueous phase have been used in the equation developed by Kobatake and co-workers to evaluate the membrane permselectivity Ps as a function of external electrolyte concentration. With the help of the equation relating Ps to phiX, the effective fixed-charge density in the membrane (where phi is a constant, 0 less than phi less than 1, and X is the membrane stochiometric charge density and can be evaluated by chemical analysis of the membrane phase), values for phiX and phi have been determined. Values of phi were low in dilute solutions and increased with increase in the concentration of the external solution. Similar behavior was noted in the case of another membrane system, cross-linked polymethacrylic acid in contact with KOH solutions. On the other hand, the membrane system, "untreated" collodion in contact with KCl solutions, exhibited a behavior in which the values of phi, low in dilute solutions, increased and then decreased following a gradual increase in the external concentration. This slight divergence in its behavior was attributed to the heterogeneity of the collodion membrane structure. The reliability of this potentiometric method to estimate effective fixed-charge density in membranes has been discussed in relation to a similar but old method due to Teorell, Meyer and Sievers. Also the significance of the values derived for phi has been pointed out.     

Plasma membrane water permeabilities of human oocytes: the temperature dependence of water movement in individual cells.

Membrane water permeability values were measured in individual fresh human pre-ovulatory oocytes using real time microscopy in a microscope diffusion chamber. The cells were exposed to anisosmotic conditions, their volume responses measured, and from these data the Lp values were computed employing the Kedem-Katchalsky analyses of irreversible thermodynamics. Lp values were measured at four temperatures for each oocyte between 37 degrees C and 10 degrees C, and the temperature-related Arrhenius activation energy (Ea) calculated. It was apparent that individual oocytes exhibited a wide range of Lp values; at 37 degrees C Lp values ranged between 0.33 and 1.80 microns/atm/min. However, each oocyte exhibited the expected inverse linear correlation between Lp and temperature, with high linear correlations (R2 values between 0.73 and 0.96). A mean value for Ea of 8.61 +/- 5.11 Kcal/mol was computed. It is apparent that pre-ovulatory human oocytes express a range of biological diversity in terms of membrane water transport, and this fact needs to be considered when attempting to formulate cryopreservation protocols for storage of these oocytes.     

Proton transport via the membrane surface

Some proton pumps, such as cytochrome c oxidase (C(c)O), translocate protons across biological membranes at a rate that considerably exceeds the rate of proton transport to the entrance of the proton-conducting channel via bulk diffusion. This effect is usually ascribed to a proton-collecting antenna surrounding the channel entrance. In this paper, we consider a realistic phenomenological model of such an antenna. In our model, a homogeneous membrane surface, which can mediate proton diffusion toward the channel entrance, is populated with protolytic groups that are in dynamic equilibrium with the solution. Equations that describe coupled surface-bulk proton diffusion are derived and analyzed. A general expression for the rate constant of proton transport via such a coupled surface-bulk diffusion mechanism is obtained. A rigorous criterion is formulated of when proton diffusion along the surface enhances the transport. The enhancement factor is found to depend on the ratio of the surface and bulk diffusional constants, pK(a) values of surface protolytic groups, and their concentration. A capture radius for a proton on the surface and an effective size of the antenna are found. The theory also predicts the effective distance that a proton can migrate on the membrane surface between a source (such as CcO) and a sink (such as ATP synthase) without fully equilibrating with the bulk. In pure aqueous solutions, protons can travel over long distances (microns). In buffered solutions, the travel distance is much shorter (nanometers); still the enhancement effect of the surface diffusion on the proton flow to a target on the surface can be tens to hundreds at physiological buffer concentrations. These results are discussed in a general context of chemiosmotic theory.     

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.     

The role of mechanical pressure difference in the generation of membrane voltage under conditions of concentration polarization

The mechanical pressure difference across the bacterial cellulose membrane located in a horizontal plane causes asymmetry of voltage measured between electrodes immersed in KCl solutions symmetrically on both sides of the membrane. For all measurements, KCl solution with lower concentration was above the membrane. In configuration of the analyzed membrane system, the concentration boundary layers (CBLs) are created only by molecular diffusion. The voltages measured in the membrane system in concentration polarization conditions were compared with suitable voltages obtained from the model of diffusion through CBLs and ion transport through the membrane. An increase of difference of mechanical pressure across the membrane directed as a difference of osmotic pressure always causes a decrease of voltage between the electrodes in the membrane system. In turn, for mechanical pressure difference across the membrane directed in an opposite direction to the difference of osmotic pressure, a peak in the voltage as a function of mechanical pressure difference is observed. An increase of osmotic pressure difference across the membrane at the initial moment causes an increase of the maximal value of the observed peak and a shift of this peak position in the direction of higher values of the mechanical pressure differences across the membrane.     

Nonlinear generalizations of the Kedem-Katachalsky equations for ionic fluxes

The transport equation of Kedem and Katchalsky for the flux of ions through a membrane is generalized to demonstrate explicitly the role of impermeant ions in determining its mathematical form. Whereas the Kedem-Katchalsky equation is linear in the salt concentrations in the bathing solutions, the more general equation is bilinear (and symmetric) in the ionic concentrations of the permeant species. The Kedem-Katchalsky flux equation is further generalized to include explicitly a term for ion-exchange in systems having more than a single permeant salt. This additional term is also bilinear (and antisymmetric) in the concentrations of the exchanging ionic species. Flux equations are derived for systems having (1) a single mono-monovalent salt, (2) two mono-monovalent salts and (3) an arbitrary number of salts with no restriction upon the valencies of the ionic components. Since it has no effect upon the form of concentration-dependent terms in the flux equations, coupling to volume flow is neglected.     

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

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

Non-steady-state O(2) diffusion in metmyoglobin solutions studied in a diffusion chamber

In this study, we studied the "passive" diffusion through myoglobin solutions by determining the oxygen diffusion coefficient (DO(2)) and the oxygen permeability (permeability O(2)) of metmyoglobin (metMb) solutions (3-33 g. 100 mL(-1)) at 25 degrees C. These oxygen diffusion parameters were determined in a diffusion chamber using a non-steady-state method and were also determined of albumin solutions (4-32 g. 100 mL(-1)) and distilled water for comparison. From these parameters, the oxygen solubility (alphaO(2)) could be calculated, because alphaO(2) = permeabilityO(2)/DO(2). Both DO(2) and permeabilityO(2) decreased with increasing metMb and albumin concentration. The values of DO(2), permeabilityO(2) and alphaO(2) of both metMb solutions and albumin solutions were comparable with literature values of methemoglobin (metHb) and serum protein solutions. The values of the metMb solutions can be used in following studies of facilitated oxygen diffusion through myoglobin solutions.     

Shape and dynamics of thermoregulating honey bee clusters

Bacterial transport systems are traditionally treated as enzymes exhibiting a saturable binding site giving rise to an apparent K(m)of transport, whereas the maximal rate of transport is regarded equivalent to the V(max)of enzymatic reactions. Thus, the Michaelis-Menten theory is usually applied in the analysis of transport data and K(m)and V(max)are derived from the treatment of data obtained from the rate of transport at varying substrate concentrations. Such an analysis tacitly assumes that the substrate recognition site of the transport system is freely accessible to substrate. However, this is not always the case. In systems endowed with high affinity in the micro M range or those recognizing large substrates or those exhibiting high V(max), the diffusion through the outer membrane may become rate determining, particularly at low external substrate concentrations. In such a situation the dependence of the overall rate of transport (from the medium into the cytoplasm) on the substrate concentration in the medium will no longer follow Michaelis-Menten kinetics. By analysing the deviation of transport data from the corresponding ideal Michaelis-Menten plot we developed a method that allows us to determine diffusion limitation through the outer membrane. The method allows us to find the correct K(m)of the transport system functioning at the inner membrane even under conditions of strong diffusion limitation through the outer membrane. The model was tested and validified with the Escherichia coli binding protein-dependent ABC transporter for maltose. The corresponding systems for sn -glycerol-3-phospate of Escherichia coli and the alpha -cyclodextrin transport of Klebsiella oxitoca were used as test systems.     

Determination of human platelet membrane permeability coefficients using the Kedem-Katchalsky formalism: estimates from two- vs three-parameter fits.

Attempts to cryopreserve human blood platelets have resulted in poor postthaw survival rates and have been inadequate for routine clinical application. As a result, most blood banks maintain platelets in nonfrozen solutions. Using this approach, platelets can be stored for only about 5 days and are then discarded. This situation greatly limits the use of platelet transfusion in clinical practice. Information regarding fundamental cryobiological characteristics can be applied to predict platelet response to cryoprotective agent (CPA) addition/removal and to cooling/warming. Methods can then be engineered to optimize cryopreservation procedures, thereby minimizing platelet damage and maximizing postthaw recovery. It was therefore the purpose of this study to determine some of the necessary biophysical parameters required for this process: (i) plasma membrane hydraulic conductivity (Lp), (ii) cryoprotectant solute permeability coefficient (Ps), (iii) the associated reflection coefficient (sigma), and (iv) their activation energies. The CPAs studied included dimethyl sulfoxide (Me2SO) and propylene glycol at 1.5 M concentration. Permeability was measured at 22, 10, and 4 degrees C using a modified Coulter counter in conjunction with a water-jacketed beaker system for temperature regulation. The Kedem-Katchalsky formalism was used to estimate the parameters using: (1) a three-parameter fit and (2) a two-parameter fit in which a noninteracting value of sigma was calculated. Two-parameter estimates were in closer agreement with previously published values, and these were used in a model to simulate addition and removal of 0.64 M (5%) and 1.0 M (7.8%) Me2SO, the most common CPA currently used in empirically determined platelet cryopreservation protocols.     

Determination of the stereoselectivity of chiral drug transport across Caco-2 cell monolayers

This study aimed to determine the transport characteristics of chiral drug enantiomers across Caco-2 cell monolayers as a model of human intestinal epithelial membrane. Esmolol was chosen as a model drug, and the study focused on the transepithelial transport of esmolol enantiomers in this in vitro model system. Separation and quantitation of (S)- and (R)-esmolol were performed by RP-HPLC with the use of GITC as a precolumn derivatizing agent. Bidirectional transport studies of 5.0-400.0 micromol/l esmolol demonstrated that the two enantiomers were transported mainly by a passive, transcellular mechanism. At concentrations of 5.0-100.0 micromol/l, enantioselective permeability of esmolol was observed. In the absorptive transport, Papp of (S)-esmolol was smaller than (R)-esmolol and vice versa for secretory transport. The enantioselectivity disappeared when the drug concentration was increased to 200.0 micromol/l. In conclusion, the transport characteristics of (S)- and (R)-esmolol were distinctly different. An enantioselective carrier-mediated mechanism in addition to passive diffusion was involved in the transport process of esmolol across Caco-2 cell monolayers.     

The multicollisional, obstructed, long-range diffusional nature of mitochondrial electron transport

Data are presented which reveal that ubiquinone (Q)-mediated electron transport is a multicollisional, obstructed, long-range diffusion process, where factors that affect the rate of lateral diffusion also affect the rate of electron transport. Based on fluorescence recovery after photobleaching measurements, it was concluded that Q-mediated electron transport occurs by the random collision of redox components which are independent lateral diffusants, each greater than 86% mobile and diffusing in a common pool. The diffusion process of Q-mediated electron transport is 1) multicollisional since the transfers of reducing equivalents between appropriate redox partners occur with less than 100% collision efficiency; 2) obstructed since its maximal rate as well as the rates of diffusion of all redox components involved vary as a function of the membrane protein density; and 3) long-range since the diffusion of all redox components is protein density-dependent, and the diffusion distance required for Q to catalyze the transfer of a reducing equivalent from Complex II to III must be, on average, greater than 37.6 nm. These findings and other theoretical treatments reveal that measurements of short-range diffusion (less than 10 nm), in which collisions between appropriate redox partners do not occur, on average, and which are not affected by membrane protein density, are irrelevant to the collisional process of electron transport. Thus, the data show that the maximum electron transport rate is dependent on both the diffusion rate and the concentration of the redox components. Sucrose was found to inhibit both the mobility of redox components as well as their electron transport rates. Data presented on the relationships between membrane viscosity, rates of lateral and rotational diffusion, and mobile fractions of redox components do not support rotationally immobile aggregates in the functional inner membrane. The high degree of unsaturated phospholipids and the absence of cholesterol in the bilayer of the native inner membrane reflect a requirement for a low resistance to motion of the redox components to compensate for the multicollisional, obstructive nature of their catalytically important collisions in this membrane. These findings support the Random Collision Model of electron transport in which the diffusion and concentration of redox components limit the maximum rate of electron transport.     

Permeability parameters of the toad isolated stratum corneum.

A technique for isolating the stratum corneum from the subjacent layers of the epithelium was developed which permits studying the stratum corneum as an isolated membrane mounted between half-chambers. The method basically consists of an osmotic shock induced by immersing a piece of skin in distilled water at 50 degrees C for 2 min. When the membrane is bathed on each surface by NaCl-Ringer's solution, its electrical resistance is 14.1 +/- 1.3 omega cm2 (n=10). This value is about 1/100 of the whole skin resistance in the presence of the same solution. The hydraulic filtration coefficient (Lp) measured by a hydrostatic pressure method, with identical solutions on each side of the membrane, is 8.8 X 10(-5) +/- 1.5 X 10(-5) cm sec-1 atm-1 (n=10) in distilled water and 9.2 X 10(-5) +/- 1.4 X 10(-5) cm sec-1 atm-1 (n=10) in NaCl-Ringer's solution. These values are not statistically different and are within the range of 1/80 to 1/120 of the whole skin Lp. The stratum corneum shows an amphoteric character when studied by KCl diffusion potentials at different pH'S. The membrane presents an isoelectric pH of 4.6 +/- 0.3 (n=10). Above the isoelectric pH the potassium transport number is higher than the chloride transport number; below it, the reverse situation is valid. Divalent cations (Ca++ or Cu++) reduce membrane ionic discrimination when the membrane is negatively charged and are ineffective when the membrane fixed charges are protonated at low pH.     

A Numerical Study of the Hydrodynamic Stable Concentration Boundary Layers in a Membrane System Under Microgravitational Conditions

On the basis of the classic formula of the concentration Rayleigh number and the Kedem–Katchalsky equation for diffusive membrane transport, we derived the equations of sixteenth order which show the dependence of the thicknesses of the concentration boundary layers on the difference of the solution concentrations, the concentration Rayleigh number, the solute permeability coefficient of the membrane and the diffusion coefficients in the solution, the kinematic viscosity of the solution, the density of solutions, the temperature and gravitational acceleration. The obtained equation has numerical solutions in the first, third and fourth quadrant of a co-ordinate system. However, only two solutions from the first quadrant of the co-ordinate system have physical meaning. Confining ourselves to the set of solutions with physical meaning only, the thicknesses of concentration boundary layers for different parameters occurring in the obtained equation were calculated numerically.     

Analysis of time hierarchy in plant cell volume changes

A simple model of plant cell volume changes is presented. It is based on Kedem-Katchalsky equations for water and solute transport and on linear approximation of the dependence of intracellular hydrostatic pressure on the cell volume. Active transport of solute is also included. The time hierarchy within the system is analyzed by appropriate normalization of variables and by the assessment of the numerical values of model coefficients. The dynamics of the system comprises a slow process of solute exchange and a fast process of water transport. This explains the wellknown biphasic response of the cell volume to a sudden change in external conditions. An approximation of equations describing the system behaviour on the basis of the Tikhonov's theorem is proposed. The approximative solution is compared with the exact numerical solution of the original equations. The approximation is very good under physiological conditions, but it ceases to hold when the solute permeability of the cell membrane increases causing the breakdown of the entire time hierarchy within the system.     

The role of cytochrome c diffusion in mitochondrial electron transport

We have compared the modes and rates of cytochrome c diffusion to the rates of cytochrome c-mediated electron transport in isolated inner membranes and in whole intact mitochondria. For inner membranes, an increasing ionic strength results in an increasing rate of cytochrome c diffusion, a decreasing concentration (affinity) of cytochrome c near the membrane surface as well as near its redox partners, and an increasing rate of electron transport. For intact mitochondria, an increasing ionic strength results in a parallel, increasing rate of cytochrome c-mediated electron transport. In both inner membranes and intact mitochondria the rate of cytochrome c-mediated electron transport is highest at physiological ionic strength (100-150 mM), where the diffusion rate of cytochrome c is highest and its diffusion mode is three-dimensional. In intact mitochondria, succinate and duroquinol-driven reduction of endogenous cytochrome c is greater than 95% at all ionic strengths, indicating that cytochrome c functions as a common pool irrespective of its diffusion mode. Using a new treatment to obtain bimolecular diffusion-controlled collision frequencies in a heterogenous diffusion system, where cytochrome c diffuses laterally, pseudo-laterally, or three-dimensionally while its redox partners diffuse laterally, we determined a high degree of collision efficiency (turnover/collisions) for cytochrome c with its redox partners for all diffusion modes of cytochrome c. At physiological ionic strength, the rapid diffusion of cytochrome c in three dimensions and its low concentration (affinity) near the surface of the inner membrane mediate the highest rate of electron transport through maximum collision efficiencies. These data reveal that the diffusion rate and concentration of cytochrome c near the surface of the inner membrane are rate-limiting for maximal (uncoupled) electron transport activity, approaching diffusion control.     

Mechanistic equations for membrane substance transport and their identity with Kedem-Katchalsky equations

Since the physical interpretation of practical Kedem-Katchalsky (KK) equations is not clear, we consider an alternative, mechanistic approach to membrane transport generated by osmotic and hydraulic pressure. We study a porous membrane with randomly distributed pore sizes (radii). We postulate that reflection coefficient (sigma(p)) of a single pore may equal 1 or 0. From this postulate we derive new (mechanistic) transport equations. Their advantage is in clear physical interpretation and since we show they are equivalent to the KK equations, the interpretation of the latter became clearer as well. Hence the equations allow clearer and more detailed interpretation of results concerning membrane substances transport.     

Proton transport by phosphate diffusion--a mechanism of facilitated CO2 transfer

We have measured CO2 fluxes across phosphate solutions at different carbonic anhydrase concentrations, bicarbonate concentration gradients, phosphate concentrations, and mobilities. Temperature was 22-25 degrees C, the pH of the phosphate solutions was 7.0-7.3. We found that under physiological conditions of pH and pCO2 a facilitated diffusion of CO2 occurs in addition to free diffusion when (a) sufficient carbonic anhydrase is present, and (b) a concentration gradient of HCO3- is established along with a pCO2 gradient, and (c) the phosphate buffer has a mobility comparable to that of bicarbonate. When the phosphate was immobilized by attaching 0.25-mm-long cellulose particles, no facilitation of CO2 diffusion was detectable. A mechanism of facilitated CO2 diffusion in phosphate solutions analogous to that in albumin solutions was proposed on the basis of these findings: bicarbonate diffusion together with a facilitated proton transport by phosphate diffusion. A mathematical model of this mechanism was formulated. The CO2 fluxed predicted by the model agree quantitatively with the experimentally determined fluxes. It is concluded that a highly effective proton transport mechanism acts in solutions of mobile phosphate buffers. By this mechanism; CO2 transfer may be increased up to fivefold and proton transfer may be increased to 10,000-fold.