The steady-state transport of oxygen through hemoglobin solutions

The steady-state transport of oxygen through hemoglobin solutions was studied to identify the mechanism of the diffusion augmentation observed at low oxygen tensions. A novel technique employing a platinum-silver oxygen electrode was developed to measure the effective diffusion coefficient of oxygen in steady-state transport. The measurements were made over a wider range of hemoglobin and oxygen concentrations than previously reported. Values of the Brownian motion diffusion coefficient of oxygen in hemoglobin solution were obtained as well as measurements of facilitated transport at low oxygen tensions. Transport rates up to ten times greater than ordinary diffusion rates were found. Predictions of oxygen flux were made assuming that the oxyhemoglobin transport coefficient was equal to the Brownian motion diffusivity which was measured in a separate set of experiments. The close correlation between prediction and experiment indicates that the diffusion of oxyhemoglobin is the mechanism by which steady-state oxygen transport is facilitated.     

Solubilities and Diffusivities of Oxygen in Hemolyzed Human Blood Solutions

By continuous absorption and by bubble collapse methods respectively, the solubilities and diffusion coefficients of oxygen in water and in dilute solutions of human hemoglobin (1.11, 2.22, and 4.44 wt%) have been determined at one atmosphere and 10°, 20°, 30°, 40°, 50°, and 60°C. Measured equilibrium constants, oxygen/hemoglobin ratios and isochoric heats of solution have been interpreted in terms of various mechanisms for oxygen-hemoglobin interaction. Oxygen diffusivities obtained experimentally for the hemolyzed blood solutions have been found to compare favorably with those predicted by a model of facilitated transport proposed by Houghton (1966). The diffusion measurements indicate that, while kinetic phenomena cannot be ignored, the over-all rate of exchange of oxygen with hemoglobin is not a controlling factor in facilitated diffusion. Anomalous equilibrium constants and temperature coefficients have been observed in the most dilute hemoglobin solution at the lowest temperatures.     

Transient effects on the initial rate of oxygenation of red blood cells

The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O₂ in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.     

Reaction rates of oxygen with hemoglobin measured by non-equilibrium facilitated oxygen diffusion through hemoglobin solutions

The purpose of this study was to verify the concept of non-equilibrium facilitated oxygen diffusion. This work succeeds our previous study, where facilitated oxygen diffusion by hemoglobin was measured at conditions of chemical equilibrium, and which yielded diffusion coefficients of hemoglobin and of oxygen. In the present work chemical non-equilibrium was induced using very thin diffusion layers. As a result, facilitation was decreased as predicted by theory. Thus, this work presents the first experimental demonstration of non-equilibrium facilitated oxygen diffusion. In addition, association and dissociation rate parameters of the reaction between oxygen and bovine and human hemoglobin were calculated and the effect of the homotropic and heterotropic interactions on each rate parameter was demonstrated. The results indicate that the homotropic interaction--which leads to increasing oxygen affinity with increasing oxygenation--is predominantly due to an increase in the association rate. The heterotropic interaction--which leads to decreasing oxygen affinity by anionic ligands--appears to be effected in two ways. Cl- increases the dissociation rate. In contrast, 2,3-diphosphoglycerate decreases the association rate.     

Facilitated diffusion: the elasticity of oxygen supply.

It is well established that the presence of oxygen-carrying proteins such as haemoglobin can facilitate the diffusion of oxygen through a solution. In this paper, it is shown that some properties of a facilitated flow are substantially different from those of unfacilitated flux, including especially the stability of the tension at which the oxygen arrives at the end of the diffusion path. The concept of the “output resistance” of the supply is introduced, and a facilitated pathway is shown to have a lowered resistance. A role for the storage capacity of the bound oxygen reservoir is also developed; it is shown that delivery oxygen tensions are stabilized against transient changes in oxygen demand. In treating the equations of facilitated diffusion, a simplified approach is used to take account of boundary layers in the solution where deviations from oxygen-protein equilibrium are significant. A measure of the thickness and importance of such boundary layers is calculated.     

The effect of auxiliary conditions on intestinal unstirred layer diffusion modelled by numerical simulation.

Estimation of intestinal unstirred layer thickness usually involves inducing transmural potential difference changes by altering the content of the solution used to perfuse the small intestine. Osmotically active solutes, such as mannitol, when added to the luminal solution diffuse across the unstirred water layer (UWL) and induce osmotically dependent changes in potential difference. As an alternative procedure, the sodium ion in the luminal fluid can be replaced by another ion. As the sodium ion diffuses out of the UWL, the change in concentration next to the intestinal membrane alters the transmural potential difference. In both cases, UWL thickness is calculated from the time course of the potential difference changes, using a solution to the diffusion equation. The diffusion equation solution which allows the calculation of intestinal unstirred layer thickness was examined by simulation, using the method of numerical solutions. This process readily allows examination of the time course of diffusion under various imposed circumstances. The existing model for diffusion across the unstirred layer is based on auxiliary conditions which are unlikely to be fulfilled in the same intestine. The present simulation additionally incorporated the effects of membrane permeability, fluid absorption and less than instantaneous bulk phase concentration change. Simulation indicated that changes within the physiologically relevant range in the chosen auxiliary conditions (with the real unstirred layer length kept constant) can alter estimates of the apparent half-time. Consequently, changes in parameters unassociated with the unstirred layer would be misconstrued as alterations in unstirred layer thickness.     

Numerical schemes for unsteady fluid flow through collapsible tubes

The study of fluid flow through compliant tubes is a fluid-structure type problem, in which a dynamic equilibrium is maintained between the fluid and the tube wall. The analogy between this flow and gas dynamics initiated the use of a number of numerical methods which were originally developed to solve compressible flow in rigid ducts. In this study we investigate the solutions obtained by applying the Lax-Wendroff and MacCormack schemes to one-dimensional incompressible flow through a straight collapsible tube. The time-evolving numerical results were compared with exact steady-state solutions. For boundary conditions which were held fixed after a prescribed rise time, the unsteady numerical solution converges to the exact steady-state solution with very good accuracy. The stability and accuracy of all the methods depend on the amount of viscous pressure loss dictated by wall friction. Flows with undamped oscillations cannot, however, be solved with these techniques.     

Approximate treatment of the formation of a difference in the concentrations of an ion at the two sides of a membrane

The recent extension of the approximation method is applied to enable us to arrive at the time course of the concentrations at both sides of a membrane. From the differential equations which govern these, the steady-state solution is obtained in terms of the parameters, which are determined by the thickness of the diffusion layers, the chemical composition and reactions, and the diffusion constant of the membrane.     

Steady-State, Hemoglobin-Facilitated O2 Transport in Human Erythrocytes

We measured the rate of oxygen transport through thin (165 µ) films of packed erythrocytes (Hb concentration = 30 g/100 ml). Under optimal conditions steady-state O₂ diffusion was nearly three times that found when the hemoglobin was prevented from acting as a carrier molecule by carbon monoxide binding or high oxygen back pressure. After each experiment we measured hemolysis and found that it averaged less than 1%. Hemolysis could not account for the facilitation, thus proving that facilitated transport of O₂ by hemoglobin can occur in red blood cells. The rate of facilitated transport was identical for Hb solutions of equal concentration to the cells. We interpret this to mean that under the conditions of our experiments the red cell membrane offers no detectable diffusion resistance to O₂ and that the mobility of Hb in intact red cells is the same as in concentrated Hb solution.     

Oxygen equilibrium of emulsified solutions of normal and sickle hemoglobin.

Emulsions containing microdroplets of concentrated solutions of normal or sickle hemoglobin dispersed in a continous oil phase have been prepared, and the aggregation of sickle hemoglobin within microdroplets in a deoxygenated emulsion demonstrated. The equilibrium oxygen saturation of hemoglobin in the emulsions has been measured as a function of the partial pressure of oxygen by a novel spectrophotometric technique which corrects for the scattering of light in the emulsion. The half-saturation oxygen pressure and cooperativity of oxygen binding are substantially greater in concentrated (27 g/dl) solutions of sickle hemoglobin than in solutions of non-aggregating hemoglobin. The shape of the oxygen equilibrium curve of concentrated sickle hemoglobin is qualitatively discussed in terms of a previously proposed model (1).     

A new method to solve a non-steady-state multispecies biofilm model

A transient multispecies model for quantifying microbial space competition in biofilm is derived from existing models, introducing a new approach to biomass detachment modelling. This model includes inert biomass, substrate diffusion and utilization rate within the biofilm and diffusional layers. It predicts the evolution of biofilm thickness, bulk substrate concentration, species distribution and substrate concentration within the biofilm. A zero-dimensional transient model is described. Its steady-state solution is used to set up initial conditions of the one-dimensional model and case computation towards steady-state solution. Some numerical tools have been developed, enabling fast computation on microcomputers. Simulations show the validity of a zero-dimensional model and perturbated systems are also simulated. Simulations with experimental data give acceptable results.     

Dynamics of oxygen unloading from sickle erythrocytes.

The objective of this work is to theoretically model oxygen unloading in sickle red cells. This has been done by combining into a single model diffusive transport mechanisms, which have been well-studied for normal red cells, and the hemoglobin polymerization process, which has been previously been studied for deoxyhemoglobin-S solutions and sickle cells in near-equilibrium situations. The resulting model equations allow us to study the important processes of oxygen delivery and polymerization simultaneously. The equations have been solved numerically by a finite-difference technique. The oxygen unloading curve for sickle erythrocytes is biphasic in nature. The rate of unloading depends in a complicated way on (a) the kinetics of hemoglobin S polymerization, (b) the kinetics of hemoglobin deoxygenation, and (c) the diffusive transport of both free oxygen and oxy-hemoglobin. These processes interact. For example, the hemoglobin S polymer interferes with the transport of both free oxygen and unpolymerized oxy-hemoglobin, and this is accounted for in the model by diffusivities which depend on the polymer and solution hemoglobin concentration. Other parameters which influence the interaction of these processes are the concentration of 2,3-diphosphoglycerate and total hemoglobin concentration. By comparing our model predictions for oxygen unloading with simpler predictions based on equilibrium oxygen affinities, we conclude that the relative rate of oxygen unloading of cells with different physical properties cannot be correctly predicted from the equilibrium affinities. To describe the unloading process, a kinetic calculation of the sort we give here is required.     

Effect of unstirred layers on binding and reaction kinetics at a membrane surface

The effects of solution unstirred layers on the time course of chemical reactions and transport processes at a membrane surface are determined. A set of equations which describes non-steady-state diffusion through an unstirred layer coupled with chemical reaction at a membrane surface or transport through a membrane is developed. A numerical solution to the equations is obtained by uncoupling diffusive and chemical processes in an iterative manner. The diffusive process is solved by the Crank-Nicolson method; the chemical process is solved by integrating the differential equations describing the kinetics. Diffusive processes in one dimension, in three dimensions, and in the presence of an arbitrary potential near the membrane surface are solved. General characteristics of the calculated reaction time course are discussed using surface binding and membrane transport examples. Small, neglected, unstirred layers are shown to sometimes yield erroneous values of rate parameters for a surface reaction and to simulate competitive reaction kinetics. Experimental approaches for measuring unstirred layer thickness are reviewed.     

Use of a membrane-bound fluorophore to characterize diffusion boundary layers around human erythrocytes.

A novel method is used to demonstrate the presence of diffusion boundary layers around erythrocytes following rapid mixing in a stopped-flow spectrophotometer and to estimate the apparent dimensions of the diffusion boundary layers. Pink erythrocyte ghosts labeled on their external surfaces with tetramethyl rhodamine isothiocyanate (TRITC) were mixed in a stopped-flow apparatus with 50 mM NaI in Ringer's solutions. I- is an effective collisional quencher of TRITC fluorescence. TRITC fluorescence after flow stopped decreased monoexponentially with time. The concentration of I- at the cell surface as a function of time was estimated from the dependence of TRITC fluorescence on I- concentration in steady-state experiments. The kinetics of the increase in I- concentration at the cell surface was fit to two diffusional models: a planar erythrocyte ghost bounded by planar diffusion boundary layer and a spherical erythrocyte surrounded by a spherical shell diffusion boundary layer. The planar model best fits the experimental data with a diffusion boundary layer 4.68 microns thick. Using the spherical model the experimental data is best fit by a 6.9 microns diffusion boundary layer.     

Rapid Method for Assessing Oxygen Consumption Rate of Cells from Transient-state Measurements of Pericellular Dissolved Oxygen Concentration

Recently we described a method for estimating the oxygen consumption rate (OCR) of cells in static culture from equilibrium measurements of dissolved oxygen concentration (dO₂), using an oxygen-sensing microplate and the steady-state solution to Fick's Law (Guarino et al. 2004). Here we describe a complementary method for estimating OCR from the transient-state rate of change of measured dO₂. Although the system is open to the atmosphere and subject to a significant lag in sensor response, the rate of change of the measured dO₂ immediately after seeding correlated directly with both cell number and steady-state OCR. This transient-state method is linear with cell number to a much higher density than is possible with the steady-state method because it derives from measurements made before diffusion limitations can be established. For a given sensor thickness, the same correlation line between the transient and non-diffusion-limited steady-state estimates of OCR was found to apply for various preparations of rat hepatocytes. The correlation slope varied predictably with sensor thickness. Thus, despite the non-idealities of this system, the initial rate measurement offers a rapid method to obtain an estimate of absolute OCR. To demonstrate the utility of this method, we purposefully treated rat hepatocytes in ways expected to change OCR. Cells deprived of oxygen by storage under several centimeters of medium showed decreases in both OCR and viability with time. Likewise, the OCR of hepatocytes exposed to the oxidative phosphorylation inhibitor rotenone decreased, whereas those exposed to the uncoupler dinoseb increased.     

Unloading oxygen in a capillary vessel under a pathological condition

In this work, we study theoretically the unloading of oxygen from a hemoglobin molecule to the wall of a typical capillary vessel, considering that the hemoglobin under pathological conditions, obeys the rheological Maxwell model. Based on recent experimental evidences in hypertension, we consider that the red blood cells (RBCs) are composed by a single continuous medium in contrast with the classical particulate or discrete RBC models, which are only valid under normal physiological conditions. The analysis considers the hemodynamic interactions between the plasma and the hemoglobin, both circulating in a long horizontal capillary. We apply numerical and analytical methods to obtain the main fluid-dynamic characteristics for both fluids in the limit of low Reynolds and Womersley numbers. A diffusion boundary layer formulation for the oxygen transport in the combined plasma-hemoglobin core region is presented. The main aspects derived are the time and spatial evolution of the membrane. The hemoglobin and plasma velocities and the pressure distributions are shown. For the oxygen unloading the results are the oxy-hemoglobin saturation, the oxygen flux and the oxygen concentration in the cell-free plasma layer. The volume fraction of red blood cells and the Strouhal number have a great influence on the hemodynamic interactions.     

Osmosis in Cortical Collecting Tubules : A Theoretical and Experimental Analysis of the Osmotic Transient Phenomenon

This paper reports a theoretical analysis of osmotic transients and an experimental evaluation both of rapid time resolution of lumen to bath osmosis and of bidirectional steady-state osmosis in isolated rabbit cortical collecting tubules exposed to antidiuretic hormone (ADH). For the case of a membrane in series with unstirred layers, there may be considerable differences between initial and steady-state osmotic flows (i.e., the osmotic transient phenomenon), because the solute concentrations at the interfaces between membrane and unstirred layers may vary with time. A numerical solution of the equation of continuity provided a means for computing these time-dependent values, and, accordingly, the variation of osmotic flow with time for a given set of parameters including: P_f (cm s^–1), the osmotic water permeability coefficient, the bulk phase solute concentrations, the unstirred layer thickness on either side of the membrane, and the fractional areas available for volume flow in the unstirred layers. The analyses provide a quantitative frame of reference for evaluating osmotic transients observed in epithelia in series with asymmetrical unstirred layers and indicate that, for such epithelia, P_f determinations from steady-state osmotic flows may result in gross underestimates of osmotic water permeability. In earlier studies, we suggested that the discrepancy between the ADH-dependent values of P_f and P_DDw (cm s^–1, diffusional water permeability coefficient) was the consequence of cellular constraints to diffusion. In the present experiments, no transients were detectable 20–30 s after initiating ADH-dependent lumen to bath osmosis; and steady-state ADH-dependent osmotic flows from bath to lumen and lumen to bath were linear and symmetrical. An evaluation of these data in terms of the analytical model indicates: First, cellular constraints to diffusion in cortical collecting tubules could be rationalized in terms of a 25-fold reduction in the area of the cell layer available for water transport, possibly due in part to transcellular shunting of osmotic flow; and second, such cellular constraints resulted in relatively small, approximately 15%, underestimates of P_f.     

Oxygen equilibrium properties of highly purified human adult hemoglobin cross-linked between 82 beta 1 and 82 beta 2 lysyl residues by bis(3,5-dibromosalicyl) fumarate

We investigated oxygen equilibrium properties of highly purified human adult hemoglobin cross-linked between lysine-82 beta 1 and lysine-82 beta 2 by a fumaryl group, which is prepared by reaction of the CO form with bis(3,5-dibromosalicyl) fumarate. The cross-linked hemoglobin preparation isolated by the previous purification method, namely, gel filtration in the presence of 1 M MgCl2 followed by ion-exchange chromatography, was found to be contaminated with about 20% of an electrophoretically silent impurity that shows remarkably high affinity for oxygen. This impurity was separated from the desired cross-linked hemoglobin by a newly developed purification method, which utilizes a difference between the authentic hemoglobin and the impurity in reactivity of the sulfhydryl groups of cysteine-93 beta toward N-ethylmaleimide under a deoxygenated condition. After this purification procedure, the oxygen equilibrium properties of purified cross-linked hemoglobin in the absence of organic phosphate became very similar to those of unmodified hemoglobin with respect to oxygen affinity, cooperativity, and the alkaline Bohr effect. The functional similarity between the cross-linked hemoglobin and unmodified hemoglobin allows us to utilize this cross-linking for preparing asymmetric hybrid hemoglobin tetramers, which are particularly useful as intermediately liganded models. Previous studies on this type of cross-linked hemoglobin should be subject to reexamination due to the considerable amount of the impurity.     

Capsule walls as barriers to oxygen availability: Implications for the development of brooded embryos by the estuarine gastropod Crepipatella dilatata (Calyptraeidae)

Encapsulation of developing embryos imposes potential restrictions, because the capsule wall must allow for adequate inward diffusion of oxygen and for increased diffusion of oxygen as metabolic demand increases with continued development. Samples of egg capsules from the gastropod Crepipatella dilatata were used to document surface characteristics, composition of the different capsule wall layers, and alterations in wall thickness during development. The diffusion coefficient and capsule wall permeability were determined experimentally for capsules containing embryos at different developmental stages. We also determined oxygen consumption rates for various embryonic stages and for nurse eggs, which provide food for embryos during development. The capsule wall of C. dilatata possesses 2 differentiated layers: the external capsular wall (ECW) and the internal capsular wall (ICW). The ECW is compact and fibrous, features that remain invariable during development, and lacks surface features that might make some portions of the capsule wall more permeable to oxygen than others. On the other hand, the ICW is initially spongy and thick, but significantly decreases in thickness over time, particularly before the embryos begin feeding on nurse eggs. Although the capsule wall is a serious barrier to diffusion, permeability to oxygen increases over time by 112% due to the dramatic thinning of the inner capsule wall layer. Nurse eggs consume oxygen but at very low rates, supporting the idea that they correspond to living embryonic cells that have stopped their development. Respiration measurements indicated that embryos are initially supplied with enough oxygen within the egg capsules to carry out the activities characteristic of embryogenesis, even though the capsular walls show their maximum thickness and lowest permeability at this time. However, as the embryo develops its velum and becomes more active, capsule wall thickness decreases and capsule permeability to oxygen increases. Correspondingly, the oxygen demands of metamorphosed but still encapsulated specimens are approximately 135% higher than those of pre-metamorphosed sibling embryos.