Bilayer permeability during phase transition as an Erlang flow of hydrophilic pores resulting from diffusion in the radius space

The formation of hydrophilic pores in a lipid bilayer during phase transition is described using the Smoluchowski equation with an additional term of the hydrophobic pore source. This term is added to account for defects in lipid packing during phase transition. We assume that the temporal sequence of the pores is a stochastic process, a non-stationary second-order Erlang flow. Flow characteristics depend on the equation solution and determine the formation times of the hydrophilic pores. The calculated distribution of the durations of intervals between hydrophilic pores is in a good agreement with experimental data published before. In the context of this model we describe the influence of poly(ethylene glycol) on the pore formation frequency.     

Influence of surface charge and transmembrane potential on rubidium-86 efflux of human red blood cells

Summary The dependence of the rate constant of Rb⁺ efflux on extracellular cation concentration was measured. At low ionic strengths Rb⁺ efflux increased strongly. Permeability coefficients were calculated from the rate constants measured, using the Goldman flux equation, with and without making allowance for surface potentials. Only when allowance was made for surface potentials and the associated differences beween ion concentrations in the bulk solutions and at the membrane surface, the permeability coefficient remained constant. Best agreement between experimental data and theoretically calculated values was obtained when an interior surface potential of – 110 mV was assumed.When the surface charge of erythrocytes is reduced by neuraminidase, the rate constants for Rb⁺ efflux decreased, indicating a significant influence of surface potential.     

QSAR study on permeability of hydrophobic compounds with artificial membranes

We previously reported a classical quantitative structure-activity relationship (QSAR) equation for permeability coefficients (P(app-pampa)) by parallel artificial membrane permeation assay (PAMPA) of structurally diverse compounds with simple physicochemical parameters, hydrophobicity at a particular pH (logP(oct) and |pK(a)-pH|), hydrogen-accepting ability (SA(HA)), and hydrogen-donating ability (SA(HD)); however, desipramine, imipramine, and testosterone, which have high logP(oct) values, were excluded from the derived QSAR equation because their measured P(app-pampa) values were lower than calculated. In this study, for further investigation of PAMPA permeability of hydrophobic compounds, we experimentally measured the P(app-pampa) of more compounds with high hydrophobicity, including several pesticides, and compared the measured P(app-pampa) values with those calculated from the QSAR equation. As a result, compounds having a calculated logP(app-pampa)>-4.5 showed lower measured logP(app-pampa) than calculated because of the barrier of the unstirred water layer and the membrane retention of hydrophobic compounds. The bilinear QSAR model explained the PAMPA permeability of the whole dataset of compounds, whether hydrophilic or hydrophobic, with the same parameters as the equation in the previous study. In addition, PAMPA permeability coefficients correlated well with Caco-2 cell permeability coefficients. Since Caco-2 cell permeability is effective for the evaluation of human oral absorption of compounds, the proposed bilinear model for PAMPA permeability could be useful for not only effective screening for several drug candidates but also the risk assessment of chemicals and agrochemicals absorbed by humans.     

Kinetic Theory Model for Ion Movement through Biological Membranes: III. Steady-State Electrical Properties with Solution Asymmetry

An electrodiffusion model for plasma membrane ion transport, which takes into account the influence of high electric field strengths and ion-membrane molecule interactions, is presented and analyzed. A generalized Nernst-Planck equation for steady-state situations is derived which has electric field-dependent mobility and diffusion coefficients. Under the assumption of a constant electric field within the membrane, this equation is integrated to give a more general form of the Goldman equation. Based on this equation numerical computations of ionic chord conductance as a function of applied electric field strength were carried out for several permeant ion concentration ratios. The model is capable of yielding significantly larger rectification ratios than is the Goldman equation. Further, high field asymptotes to the current vs. electric field strength curve do not generally intersect at the origin.     

Leak Current Rectification in Myxicola Giant Axons : Constant field and constant conductance components

Early leak current, i.e. for times similar to the time to peak of the transient current was measured in Myxicola giant axons in the presence of tetrodotoxin. The leak current-voltage relation rectifies, showing more current for strong depolarizing pulses than expected from symmetry around the holding potential. A satisfactory practical approximation for most leak corrections is constant resting conductance. The leak current-voltage curve rectifies less than expected from the constant field equation. These curves cannot be reconstructed by summing the constant field currents for sodium and potassium using a P_Na/P_K ratio obtained in the usual way, from zero current constant field fits to resting membrane potential data. Nor can they be reconstructed by summing the constant field current for potassium with that for any other single ion. They can be reconstructed, however, by summing the constant field current for potassium with a constant conductance component. It is concluded that the leak current and the resting membrane potential, therefore, are determined by multiple ionic components, at least three and possibly many. Arguments are presented suggesting that ion permeability ratios obtained in the usual way, by fitting the constant field equation to resting membrane potential data should be viewed with skepticism.     

Normalization in the fitting of data by iterative methods. Application to tracer kinetics and enzyme kinetics

1. The normalization of biochemical data to weight them appropriately for parameter estimation is considered, with reference particularly to data from tracer kinetics and enzyme kinetics. If the data are in replicate, it is recommended that the sum of squared deviations for each experimental variable at each time or concentration point is divided by the local variance at that point. 2. If there is only one observation for each variable at each sampling point, normalization may still be required if the observations cover more than one order of magnitude, but there is no absolute criterion for judging the effect of the weighting that is produced. The goodness of fit that is produced by minimizing the weighted sum of squares of deviations must be judged subjectively. It is suggested that the goodness of fit may be regarded as satisfactory if the data points are distributed uniformly on either side of the fitted curve. A chi-square test may be used to decide whether the distribution is abnormal. The proportion of the residual variance associated with points on one or other side of the fitted curve may also be taken into account, because this gives an indication of the sensitivity of the residual variance to movement of the curve away from particular data points. These criteria for judging the effect of weighting are only valid if the model equation may reasonably be expected to apply to all the data points. 3. On this basis, normalizing by dividing the deviation for each data point by the experimental observation or by the equivalent value calculated by the model equation may both be shown to produce a consistent bias for numerically small observations, the former biasing the curve towards the smallest observations, the latter tending to produce a curve that is above the numerically smaller data points. It was found that dividing each deviation by the mean of observed and calculated variable appropriate to it produces a weighting that is fairly free from bias as judged by the criteria mentioned above. This normalization factor was tested on published data from both tracer kinetics and enzyme kinetics.     

Rubidium and cesium fluxes in muscle as related to the membrane potential

The reduction of membrane potential in frog sartorius muscle produced by rubidium and cesium ions has been studied over a wide concentration range and compared with depolarization occasioned by potassium ions. The constant field theory of passive flux has been used to predict the potential changes observed. The potential data suggest certain permeability coefficient ratios and these are compared with ratios obtained from flux data using radioactive tracers. The agreement of the flux with the potential data is good if account is taken of the inhibition of potassium flux which occurs in the presence of rubidium and cesium ions. A high temperature dependence has been observed for cesium influx (Q₁₀ = 2.5) which is correlated with the observation that cesium ions depolarize very little at low temperatures. The observations suggest that cesium ions behave more like sodium ions at low temperatures and more like potassium ions at room temperature with respect to their effect on the muscle cell resting potential. The constant field theory of passive ion flux appears to be in general agreement with the experimental results observed if account is taken of the dependence of permeability coefficients on the concentrations of ions used and of possible interactions between the permeabilities of ions.     

Application of the Smoluchowski equation with a source term to the model of lipid pore formation during a phase transition

The Smoluchowski equation, which describes pore diffusion in the radius space, with a source term, is used in modeling the process of the formation of a hydrophilic pore in a lipid bilayer during phase transition. The introduction of a hydrophobic-pore source term into the equation reflects the emergence of additional defects in a bilayer caused by the decrease in the molecule area under the transition from the liquid crystalline to the gel phase. The distribution of the time probability density calculated within the model that is required for the formation of a hydrophilic pore is in good agreement with the previously published experimental data.     

Frictional coefficients of multisubunit structures. II. Application to proteins and viruses

The theory of Kirkwood for the translational frictional coefficients of structures composed of identical subunits has been extended in the previous paper to the case where nonidentical subunits are involved. In this paper, the theory is applied to particular proteins and viruses. It is found that the calculated sedimentation coefficients of various states of aggregation of the reversibly associating proteins hemocyanin and phycocyanin are in excellent agreement with experiment. The dimensions of the fibrinogen molecule obtained from electron micrographs do not give good agreement between calculated and experimental frictional coefficients. The frictional coefficient of tobacco mosaic virus calculated without explicit consideration of end effects is in good agreement with experiment if a hydrodynamic diameter of 18O A., corresponding to the maximum diameter from x-ray studies, is used. Agreement is also good for the fast sedimenting form of bacteriophage T2 and the protein ghosts of bacteriophage λ but the slow form of T2 and whole λ have frictional coefficients considerably in excess of those calculated. Tail fiber configuration or head porosity are unable to account for the difference in sedimentation coefficients between the fast and slow forms of T2.     

Predicting permeability of regular tissue engineering scaffolds: scaling analysis of pore architecture,scaffold length,and fluid flow rate effects

The main aim of this research is to numerically obtain the permeability coefficient in the cylindrical scaffolds. For this purpose, a mathematical analysis was performed to derive an equation for desired porosity in terms of morphological parameters. Then, the considered cylindrical geometries were modeled and the permeability coefficient was calculated according to the velocity and pressure drop values based on the Darcy’s law. In order to validate the accuracy of the present numerical solution, the obtained permeability coefficient was compared with the published experimental data. It was observed that this model can predict permeability with the utmost accuracy. Then, the effect of geometrical parameters including porosity, scaffold pore structure, unit cell size, and length of the scaffolds as well as entrance mass flow rate on the permeability of porous structures was studied. Furthermore, a parametric study with scaling laws analysis of sample length and mass flow rate effects on the permeability showed good fit to the obtained data. It can be concluded that the sensitivity of permeability is more noticeable at higher porosities. The present approach can be used to characterize and optimize the scaffold microstructure due to the necessity of cell growth and transferring considerations.     

Determination of reflection coefficients of liposomes for some non-electrolytes by osmotic pressure measurement

The reflection coefficients of bilayer lipid vesicles (liposomes) of various compositions have been determined for a number of non-electrolytes. The solutes were the same and the method of measurement was essentially the same as those which have been used to estimate an equivalent pore radius for erythrocytes. The method involves matching the osmotic pressure of solutions of a permeant test solute with that of a known inpermeant solute. Reflection coefficients for cholesterol-containing liposomes and those of erythrocytes are, when account is taken of those solutes known to permeate the erythrocyte by specialized pathways, not greatly different. Lipid bilayers can thus account for most of the permeability characteristics of the cell originally interpreted as due to aqueous pores. Reflection coefficients are significantly higher for egg phosphatidylcholine membranes that contain cholesterol than those which do not. There is a strong correlation between relative permeabilities derived from reflection coefficients and oil-water partition coefficients. There is also good agreement between these permeabilities and permeabilities measured by others, which exhibit an inverse dependence on molecular size. It is suggested that this tendency of membranes to pass small molecules more readily than large molecules, other properties being equal, is a consequence of the surface pressure of the constituent monolayers of the membrane.     

Improving the damage accumulation in a biomechanical bone remodelling model

We extend, reformulate and analyse a phenomenological model for bone remodelling. The original macrobiomechanical model (MBM), proposed by Hazelwood et al. [J Biomech 2001; 34:299–308], couples a population equation for the cellular activities of the basic multicellular units (BMUs) in the bone and a rate equation to account for microdamage and repair. We propose to account for bone failure under severe overstressing by incorporating a Paris-like power-law damage accumulation term. The extended model agrees with the Hazelwood et al. predictions when the bone is under-stressed, and allows for suitably loaded bones to fail, in agreement with other MBM and experimental data regarding damage by fatigue. We numerically solve the extended model using a convergent algorithm and show that for unchanging loads, the stationary solution captures fully the model behaviour. We compute and analyse the stationary solutions. Our analysis helps guide additional extensions to this and other BMU activity based models.     

Correlation of the turnover number of the ATP synthase in liposomes with the proton flux and the proton potential across the membrane

The fluorescent indicator pyranine was used for recording the internal pH of liposomes. The proton permeability was deduced from the velocity of the internal pH increase which was caused by shifting the external pH from 7 to 9. From valinomycin titration of the proton permeability in the presence of internal and external KCl (0.1 M), the permeability coefficient of H+ (PH) was obtained as 10(-4) cm/s at 22 degrees C. The coefficient was twice this value with the ATP synthase isolated from Wolinella succinogenes present in the liposomal membrane (10 mg protein/g phospholipid). ADP and phosphate had no effect on the latter PH. The protonophore TTFB (5 mumol/g phospholipid) increased the PH by 3 orders of magnitude. The permeability coefficients of H+ and K+ were used for calculating the delta uH and the proton flux associated with the phosphorylation which was driven by gradients of H+ and K+. For the conditions of limiting permeability of K+, the following conclusions were drawn. (1) In the steady state of rapid ion flux, the electrical potential across the liposomal membrane as calculated according to the Goldman equation, is directed opposite to the corresponding Nernst potential which is calculated from the K+ gradient. (2) The maximum turnover numbers of phosphorylation require a delta uH of 200-220 mV across the liposomal membrane. These values of delta uH and the corresponding turnover numbers are close to those brought about by the bacterial electron transport and the coupled phosphorylation. (3) The velocity of phosphorylation is linearly related to the proton flux. The slope of the line can be explained on the basis of an H+/ATP ratio of approx. 3.     

Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids

In order to determine the role of micromilieu in tumour spheroid growth, a mathematical model was developed to predict EMT6/Ro spheroid growth and microenvironment based upon numerical solution of the diffusion/reaction equation for oxygen, glucose, lactate ion, carbon dioxide, bicarbonate ion, chlorine ion and hydrogen ion along with the equation of electroneutrality. This model takes into account the effects of oxygen concentration, glucose concentration and extracellular pH on cell growth and metabolism. Since independent measurements of EMT6/Ro single cell growth and metabolic rates, spheroid diffusion constants, and spinner flask mass transfer coefficients are available, model predictions using these parameters were compared with published data on EMT6/Ro spheroid growth and micro-environment. The model predictions of reduced spheroid growth due to reduced cell growth rates and cell shedding fit experimental spheroid growth data below 700 microns, but overestimated the spheroid growth rate at larger diameters. Predicted viable rim thicknesses based on predicted near zero glucose concentrations fit published viable rim thickness data for 1000 microns spheroids grown at medium glucose concentrations of 5.5 mM or less. However, the model did not accurately predict the onset of necrosis. Moreover, the model could not predict the observed decreases in oxygen and glucose metabolism seen in spheroids with time, nor could it predict the observed growth plateau. This suggests that other unknown factors, such as inhibitors or cell-cell contact effects, must also be important in affecting spheroid growth and cellular metabolism.     

Effects of ultraviolet radiation on the membranes ofChara corallina

Summary The effects of 253.7 nm ultraviolet (UV) radiation on the membrane properties ofChara corallina have been studied. UV irradiation caused depolarization of the membrane potential (p.d.) and a decrease in membrane resistance. These effects were largely reversible with steady values being obtained within 40 minutes after the UV was turned off. The effects on ionic fluxes of Na⁺, K⁺ and Cl^– have also been studied using radioactive tracer techniques. The influxes were unchanged by irradiation. The chloride efflux was increased sevenfold during the irradiation period but recovered to the pre-irradiation value within 30 minutes after the irradiation period. The potassium efflux was also increased and reached a maximum 10 minutes after irradiation. The resting potential and the average depolarized p.d. reached during irradiation were in good agreement with those calculated from permeability coefficients indicated by the observed passive fluxes, using the Goldman equation for p.d. However, the plasmalemma resistance and its change due to irradiation did not match the values calculated from the same permeability coefficients used to estimate p.d. This disagreement, and an apparent imbalance in the charge transferred across the resting or irradiated plasmalemma, suggest the participation of another ion species as well as K⁺, Na⁺ and Cl^–.     

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.     

Partitioning and diffusion of large molecules in fibrous structures

This paper consists of three parts. In the first part we present diffusion and partition coefficients of proteins in agarose that were measured with gel permeation chromatography. In the second part we present a partition model which includes the effects of solute concentration and of cosolutes. In the third part we modify and extend Ogston’s diffusion equation to account for the effects of solute and fiber flexibility, solute concentration and cosolutes. We find good agreement between the proposed models and data from the literature.     

Thermodynamic modeling of the partitioning of biomolecules in aqueous two-phase systems using a modified Flory–Huggins equation

A modified Flory–Huggins equation accounting for the solvation of polymer molecules by water molecules was used to model the phase behavior of aqueous two-phase systems (ATPS) formed by poly(ethylene glycol) (PEG) and dextran. The parameters of the equation were obtained by fitting experimental equilibrium data either accounting for or disregarding dextran polidispersity. The modified equation was subsequently applied to calculate partition coefficients of biomolecules in these systems. It was found that accounting for polidispersity did not affect significantly the calculated phase equilibrium, but increased the agreement of calculated partition coefficients with experimental data. Further improvement was obtained by using a size dependent interaction parameter for dextran pseudo-components.     

The reliability of relative anion-cation permeabilities deduced from reversal (Dilution) potential measurements in ion channel studies

Measurements of anion-cation permeability ratios (e.g., P _Cl/P _Na) are most readily made by measuring changes in zero-current reversal potential when the salt concentration on one side of the membrane (e.g., external NaCl) is decreased. This is particularly useful for measuring changes in ion selectivity in wild-type and mutant channels, such as those of the ligand-gated ion channel superfamily, and has shown that many of these channels have a significant permeability to counter-ions. One Brownian dynamics study of ion permeation through such narrow ion channels failed to observe such counter-ion movement, although later, another Brownian dynamics study did observe counter-ion movement through simulations of the same channels. The question has been raised as to the reliability of such reversal potential measurements for determining permeability ratios, particularly given the use of an equation such as the Goldman-Hodgkin-Katz (GHK) equation, which is often used to calculate such ratios. A new derivation of the GHK equation in terms of activity coefficients is also included. The application of irreversible thermodynamics will be shown to qualitatively support the reliability of such experimental anion-cation permeability values derived from reversal potential measurements. It will then be shown that for such zero-current situations, different electrodiffusion models, with very different underlying assumptions, produce almost identical relative permeabilities (and reversal potentials). Finally, the results of the two Brownian dynamics simulation studies and the relationship between reversal potentials and relative permeability will be discussed.     

Failure of the Nernst-Einstein Equation to Correlate Electrical Resistances and Rates of Ionic Self-Exchange across Certain Fixed Charge Membranes

The electrical resistances and rates of self-exchange of univalent critical ions across several types of collodion matrix membranes of high ionic selectivity were studied over a wide range of conditions. The relationship which was observed between these quantities with membranes of a certain type, namely those activated with poly-2-vinyl-N-methyl pyridinium bromide, cannot be explained on the basis of current concepts of the movement of ions across ion exchange membranes. Rates of self-exchange across these membranes were several times greater than those calculated from the electrical resistances of the membranes on the basis of an expression derived by the use of the Nernst-Einstein equation. The magnitude of the discrepancy was greatest at low concentrations of the ambient electrolyte solution and was independent of the species of both critical and noncritical ions. The data obtained with other types of collodion matrix membranes were, at least approximately, in agreement with the predictions based on the Nernst-Einstein equation. Self-exchange rates across the anion permeable protamine collodion membranes, and across the cation permeable polystyrene sulfonic acid collodion membranes, were about 20% less than those calculated from the electrical resistances. The direction and magnitude of these differences, also observed by other investigators, are qualitatively understood as an electroosmotic effect. With cation permeable membranes prepared by the oxidation of preformed collodion membranes, almost exact agreement was obtained between measured and calculated self-exchange rates; the cause of the apparent absence of an electroosmotic effect with these membranes is unknown.