Membrane Transport Generated by the Osmotic and Hydrostatic Pressure. Correlation Relation for Parameters L p, σ, and ω

Standard approach to membrane transport generated by osmotic andhydrostatic pressures, developed by Kedem and Katchalsky, is based onprinciples of thermodynamics of irreversible processes. In this paper wepropose an alternative technique. We derive transport equations from fewfairly natural assumptions and a mechanistic interpretation of the flows.In particular we postulate that a sieve-type membrane permeability isdetermined by the pore sizes and these are random within certain range.Assuming that an individual pore is either permeable or impermeable tosolute molecules, the membrane reflection coefficient depends on the ratioof permeable and impermeable pores. Considering flows through permeableand impermeable pores separately, we derive equations for the total volumeflux, solute flux and the solvent flux across the membrane. Comparing themechanistic equations to the Kedem-Katchalsky equations we find the formereasier to interpret physically. Based on the mechanistic equations we alsoderive a correlation relation for the membrane transport parameters L _p,, and . This relation eliminates the need for experimentaldetermination of all three phenomenological parameters, which in somecases met with considerable difficulties.     

Application of irreversible thermodynamics to a functional description of the tumor cell membrane

The permeability of the membranes of several strains of Ehrlich mouse ascites tumor cells to a series of homologous glycols was analyzed with equations from irreversible thermodynamics. Experimental data were fitted to the appropriate equations of Kedem and Katchalsky by means of an analog computer and values for σ, L_p, and ωRT were calculated. These parameters would become part of a functional profile of the membrane of the tumor cell. The glycols cross the membrane with energies of activation ranging from 16 kcal/mole for ethylene glycol to 20 kcal/mole for triethylene glycol. Entropies of activation ranged from 17 to 27 entropy units. This analysis provided several additional dividends. It is one of the few examples with a mammalian cell where the kinetics of non-electrolyte transport have been described completely by the equations of irreversible thermodynamics. It has provided the means to distinguish how much of the transport occurred through porous paths in the membrane and how much involved solution within the matrix of the membrane. Finally, several tumor cell populations have been distinguished through these functional properties of their membranes.     

Determination of Urea Permeability in Red Cells by Minimum Method : A test of the phenomenological equations

A new method has been developed for measuring the permeability coefficient, ω, of small nonelectrolytes. The method depends upon a mathematical analysis of the time course of cell volume changes in the neighborhood of the minimum volume following addition of a permeating solute to an isosmolal buffer. Coefficients determined by the minimum volume method agree with those obtained using radioactive tracers. ω for urea in human red cells was found to decrease as the volume flow, J_v, into the cell increased. Such behavior is entirely unexpected for a single uniform rate-limiting barrier on the basis of the linear phenomenological equations derived from irreversible thermodynamics. However, the present findings are consonant with a complex membrane system consisting of a tight barrier on the outer face of the human red cell membrane and a somewhat less restrictive barrier behind it closer to the inner membrane face. A theoretical analysis of such a series model has been made which makes predictions consistent with the experimental findings.     

Properties of membrane stationary states

Summary The flux of permeant species through a membrane is examined using discrete state stochastic models for the transport process within the membrane. While a membrane flux is maintained due to a concentration gradient between bathing solutions, the distribution of species within the membrane evolves to a time invariant configuration which can differ significantly from the equilibrium configuration. Some special properties of these stationary states are examined using linear, microcanonical models for the membrane transport process. Analysis of these models reveals properties which are masked by the phenomenological analysis of irreversible thermodynamics. For example, the models can be used to study the nature of multi-state relaxation within the membrane by observation of the time dependence of the net membrane flux when the membrane is perturbed from its stationary state distribution. Under some conditions, multi-state models will produce relaxation similar to that observed for single-state processes. The symmetry within the membrane is a critical factor for monitoring relaxation processes within the membrane. Because of the stationary nature of the membrane configuration, statistical thermodynamic variables can be defined for the membrane configuration. The total system is not in equilibrium since the baths must still be described by dissipation functions. In the stationary state, the configurational entropy of the membrane is lowered relative to equilibrium and is shown to depend quadratically on the time independent parameter (j/p) wherej is the membrane flux andp is a characteristic transition probability for intra-membrane transitions. The basic membrane system serves as a quantitative example of the entropy reduction possible in a stationary state system. An allosteric transition mediated by the stationary state configuration is examined as a means of utilizing this negentropy production.     

Osmotic Transport across Cell Membranes in Nondilute Solutions: A New Nondilute Solute Transport Equation

The fundamental physical mechanisms of water and solute transport across cell membranes have long been studied in the field of cell membrane biophysics. Cryobiology is a discipline that requires an understanding of osmotic transport across cell membranes under nondilute solution conditions, yet many of the currently-used transport formalisms make limiting dilute solution assumptions. While dilute solution assumptions are often appropriate under physiological conditions, they are rarely appropriate in cryobiology. The first objective of this article is to review commonly-used transport equations, and the explicit and implicit assumptions made when using the two-parameter and the Kedem-Katchalsky formalisms. The second objective of this article is to describe a set of transport equations that do not make the previous dilute solution or near-equilibrium assumptions. Specifically, a new nondilute solute transport equation is presented. Such nondilute equations are applicable to many fields including cryobiology where dilute solution conditions are not often met. An illustrative example is provided. Utilizing suitable transport equations that fit for two permeability coefficients, fits were as good as with the previous three-parameter model (which includes the reflection coefficient, σ). There is less unexpected concentration dependence with the nondilute transport equations, suggesting that some of the unexpected concentration dependence of permeability is due to the use of inappropriate transport equations.     

Red blood cell membrane water permeability increases with length of ex vivo storage

Water transport across the red blood cell (RBC) membrane is an essential cell function that needs to be preserved during ex vivo storage. Progressive biochemical depletion during storage can result in significant conformational and compositional changes to the membrane. Characterizing the changes to RBC water permeability can help in evaluating the quality of stored blood products and aid in the development of improved methods for the cryopreservation of red blood cells. This study aimed to characterize the water permeability (L_p), osmotically inactive fraction (b), and Arrhenius activation energy (E_a) at defined storage time-points throughout storage and to correlate the observed results with other in vitro RBC quality parameters. RBCs were collected from age- and sex-matched blood donors. A stopped flow spectrophotometer was used to determine L_p and b by monitoring changes in hemoglobin autofluorescence when RBCs were exposed to anisotonic solutions. Experimental values of L_p were characterized at three different temperatures (4, 20 and 37 °C) to determine the E_a. Results showed that L_p, b, and E_a of stored RBCs significantly increase by day 21 of storage. Degradation of the RBC membrane with length of storage was seen as an increase in hemolysis and supernatant potassium, and a decrease in deformability, mean corpuscular hemoglobin concentration and supernatant sodium. RBC osmotic characteristics were shown to change with storage and correlate with changes in RBC membrane quality metrics. Monitoring water parameters is a predictor of membrane damage and loss of membrane integrity in ex vivo stored RBCs.     

Membrane Permeability Modeling: Kedem–Katchalsky vs a Two-Parameter Formalism

The analysis of experiments for the purpose of determining cell membrane permeability parameters is often done using the Kedem–Katchalsky (KK) formalism (1958). In this formalism, three parameters, the hydraulic conductivity (L_p), the solute permeability (P_s), and a reflection coefficient (ς), are used to characterize the membrane. Sigma was introduced to characterize flux interactions when water and solute (cryoprotectant) cross the membrane through a common channel. However, the recent discovery and characterization of water channels (aquaporins) in biological membranes reveals that aquaporins are highly selective for water and do not typically cotransport cryoprotectants. In this circumstance, sigma is a superfluous parameter, as pointed out by Kedem and Katchalsky. When sigma is unneeded, a two-parameter model (2P) utilizing onlyL_pandP_sis sufficient, simpler to implement, and less prone to spurious results. In this paper we demonstrate that the 2P and KK formalism yield essentially the same result (L_pandP_s) when cotransporting channels are absent. This demonstration is accomplished using simulation techniques to compare the transport response of a model cell using a KK or 2P formalism. Sigma is often misunderstood, even when its use is appropriate. It is discussed extensively here and several simulations are used to illustrate and clarify its meaning. We also discuss the phenomenological nature of transport parameters in many experiments, especially when both bilayer and channel transport are present.     

A Phenomenological Theory of Muscular Contraction: II. Generalized Length Variations

In part I of this series, the theory of irreversible thermodynamics was applied to the sliding filament model to obtain rate equations for a contracting muscle at the in situ length l_o. In this paper we extend the theory to include length variations derived from the sliding filament model of contracting muscle using the work of Gordon, Huxley, and Julian (1). Accepting the validity of Hill's forcevelocity relation (2) at the in situ length, we show that Hill's equation is valid for any length provided that the values of the parameters, a, b, and V_m vary with length as derived herein. The predicted variation with length of the velocity for a lightly loaded isotonic contraction is shown to agree well with that measured by Gordon, Huxley, and Julian (1). Chemical rates are derived as functions of length using parameters that can be obtained experimentally.     

A Physical Interpretation of the Phenomenological Coefficients of Membrane Permeability

A "translation" of the phenomenological permeability coefficients into friction and distribution coefficients amenable to physical interpretation is presented. Expressions are obtained for the solute permeability coefficient ω and the reflection coefficient σ for both non-electrolytic and electrolytic permeants. An analysis of the coefficients is given for loose membranes as well as for dense natural membranes where transport may go through capillaries or by solution in the lipoid parts of the membrane. Water diffusion and filtration and the relation between these and capillary pore radius of the membrane are discussed. For the permeation of ions through the charged membranes equations are developed for the case of zero electrical current in the membrane. The correlation of σ with ω and L_p for electrolytes resembles that for non-electrolytes. In this case ω and σ depend markedly on ion concentration and on the charge density of the membrane. The reflection coefficient may assume negative values indicating anomalous osmosis. An analysis of the phenomena of anomalous osmosis was carried out for the model of Teorell and Meyer and Sievers and the results agree with the experimental data of Loeb and of Grim and Sollner. A set of equations and reference curves are presented for the evaluation of ω and σ in the transport of polyvalent ions through charged membranes.     

The metrical structure of aging (dissipative) systems

A quantification of the aging of a system is achieved by establishing a metric algebra based upon the dissipation function associated with the system. The phenomenological coefficient, L_ij, of irreversible thermodynamics is shown to be a dynamical analogue of the metric tensor, g_ij, of geometry. Given this metric for aging systems, it then becomes possible to compare the aging of two similar systems exposed to different environmental forces and to use the concept of age-preserving transformations to determine under what conditions two different systems will age at the same rate.     

Laser interferometric investigation of solute transport through membrane-concentration boundary layer system

We investigate diffusive transport in a membrane system with a horizontally mounted membrane under concentration polarization conditions performed by a laser interferometry method. The data obtained from two different theoretical models are compared to the experimental results of the substance flux. In the first model, the membrane is considered as infinitely thin, while in the second one as a wall of finite thickness. The theoretical calculations show sufficient correspondence with the experimental results. On the basis of interferometric measurements, the relative permeability coefficient (ζ_s) for the system, consisting of the membrane and concentration boundary layers, was also obtained. This coefficient reflects the concentration polarization of the membrane system. The obtained results indicate that the coefficient ζ_s of the membrane-concentration boundary layer system decreases in time and seems to be independent of the initial concentration of the solute.     

Theoretical analysis of Lumry-Eyring models in differential scanning calorimetry

A theoretical analysis of several protein denaturation models (Lumry-Eyring models) that include a rate-limited step leading to an irreversibly denatured state of the protein (the final state) has been carried out. The differential scanning calorimetry transitions predicted for these models can be broadly classified into four groups: situations A, B, C, and C′. (A) The transition is calorimetrically irreversible but the rate-limited, irreversible step takes place with significant rate only at temperatures slightly above those corresponding to the transition. Equilibrium thermodynamics analysis is permissible. (B) The transition is distorted by the occurrence of the rate-limited step; nevertheless, it contains thermodynamic information about the reversible unfolding of the protein, which could be obtained upon the appropriate data treatment. (C) The heat absorption is entirely determined by the kinetics of formation of the final state and no thermodynamic information can be extracted from the calorimetric transition; the rate-determining step is the irreversible process itself. (C′) same as C, but, in this case, the rate-determining step is a previous step in the unfolding pathway. It is shown that ligand and protein concentration effects on transitions corresponding to situation C (strongly rate-limited transitions) are similar to those predicted by equilibrium thermodynamics for simple reversible unfolding models. It has been widely held in recent literature that experimentally observed ligand and protein concentration effects support the applicability of equilibrium thermodynamics to irreversible protein denaturation. The theoretical analysis reported here disfavors this claim.     

A parallel path model forNecturus proximal tubule

Summary A parallel path model based on the principles of nonequilibrium thermodynamics was developed for theNecturus proximal tubule. The cellular path was represented as a luminal membrane followed by an irreversible active NaCl transport system in the peritubular barrier. The shunt pathway was described as three coarse barriers in series: tight junction, lateral intercellular spaces, and basement membrane with connective tissue. Volume and solute flows were predicted by the model equations as a function of applied electric current. Variations of the model parameters revealed the quantitative importance of the shunt path properties and the relative insensitivity of epithelial transport to changes in most cell parameters. Circulation of electric current and solute within the epithelium were shown to significantly influence the bahavior of the tubule in the presence of an electric field. Values for all transport parameters of the shunt path and epithelium were calculated and compared with available experimental evidence. Volume flow and electric currents predicted by the model compared favorably with experimental observations.     

Capillary operators

The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL _p[0,∞], (1<-p<-∞). A discussion is included of theL _p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O _K (relating extravascular concentration to intravascular) and K _{a, k} (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O _K ‖ _p <-1, ‖K _{a, k} ‖ _p <-1), isotone, linear operators onL _p . The uniform convergence relation $$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$ (as operators onL _p) is derived, whereP _n (a) is the Poisson probabilitye ^?a a ⁿ /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖O_k‖=‖K_a,k‖_p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.     

Thermal membrane potential across charged membranes in NaCl-NH4Cl and LiCl-NH4Cl solutions

Measurements of the thermal membrane potential across cation exchange membranes were carried out by using aqueous solutions containing two 1-1 electrolytes, with an anion in common. The same solution was used on both sides of the membrane. In all cases a good linear relationship was observed between the thermal membrane potential Δψ and the temperature difference ΔT (in the range ΔT = ± 10°C). Assuming that the activity of one cation is equal to that of another cation in the solutions and the sum of transport numbers of cations is unity, the plot of Δψ/ΔT vs logarithmic activity of one cation is linear with a slope of R/F. These experimental results aie in agreement with a theory presented previously. From the analysis of thermal membrane potential in mixtures of electrolytes it is obtained that the cross coefficient of cation-cation interaction in membranes is negative and about 6 to 9% of the main coefficient.     

Maximum sustainable yield estimates of spiny lobster fishery in Pakistan using non-equilibrium CEDA package

Maximum sustainable yield estimates of spiny lobster fishery were analyzed using catch effort data analysis (CEDA) computer programme. The major parameters of this package are: Maximum sustainable yield (MSY), catchability coefficient (q) carrying capacity (K), intrinsic growth rate (r), Replacement yield and Final population. CEDA has ability to assess the parameters of Fox, Schaefer and Pella-Tomlinson models. In addition it has an ability to estimate three error assumptions i.e. normal, log normal and gamma. In this study, the Maximum sustainable yield outputs of three models of Fox, Schaefer and Pella-Tomlinson are: 828 t, 970 t and 970 t respectively. The outputs of error assumption of normal and log normal are 983 t (R ² = 0.57) and 950 t (R ² = 0.53) in Schaefer and Pella-Tomlinson models respectively. MSY outputs of normal error assumption of Fox are 817 t (R ² = 0.56). All the gamma error assumptions are (790 t) similar. The coefficient of variation (cv) of the estimated MSY was about 0.7 and the larger value (1.0) whereas lowest (0.5) were recorded. The Fox model output are more conservative hence the best fits and is close to the annual average landings of the spiny lobster fishery in Pakistan.