1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Appendix

We have developed an experimental approach that allows us to quantify unstirred layers around cells suspended in stirred solutions. This technique is applicable to all types of transport measurements and was applied here to the 18O technique used to measure CO2 permeability of red cells inline image. We measure inline image in well-stirred red cell (RBC) suspensions of various viscosities adjusted by adding different amounts of 60 kDa dextran. Plotting inline imagevs. viscosity ν gives a linear relation, which can be extrapolated to ν= 0. Theoretical hydrodynamics predicts that extracellular unstirred layers vanish at zero viscosity when stirring is maintained, and thus this extrapolation gives us an estimate of the inline image free from extracellular unstirred layer artifacts. The extrapolated value is found to be 0.16 cm s−1 instead of the experimental value in saline of 0.12 cm s−1 (+30%). This effect corresponds to an unstirred layer thickness of 0.5 μm. In addition, we present a theoretical approach modelling the actual geometrical and physico-chemical conditions of 18O exchange in our experiments. It confirms the role of an extracellular unstirred layer in the determination of inline image. Also, it allows us to quantify the contribution of the so-called intracellular unstirred layer, which results from the fact that in these transport measurements – as in all such measurements in general – the intracellular space is not stirred. The apparent thickness of this intracellular unstirred layer is about 1/4–1/3 of the maximal intracellular diffusion distance, and correction for it results in a true inline image of the RBC membrane of 0.20 cm s−1. Thus, the order of magnitude of this inline image is unaltered compared to our previous reports. Discussion of the available evidence in the light of these results confirms that CO2 channels exist in red cell and other membranes, and that inline image of red cell membranes in the absence of these channels is quite low.

It has long been recognized (Gutknecht et al. 1977; Holland et al. 1985; Wunder et al. 1997; Wunder & Gros, 1998; Endeward & Gros, 2005; Endeward et al. 2006b; Missner et al. 2008) that unstirred layers on cellular surfaces constitute a special problem when the transport of gases across cell membranes is measured. This is due to the fact that in many, although not all, cell membranes the permeability to gases is several orders of magnitude greater than that of ions. This holds even in the case of ions that are as permeable as bicarbonate in the red blood cell membrane. On the other hand, the diffusion coefficients of CO2 and other gases in water are not much greater than those of ions (Gros et al. 1976). While in the case of ions the diffusion resistance of an unstirred layer is often negligible in comparison to the diffusion resistance of the membrane itself, this implies that in the case of CO2 and other gases the diffusion resistance exhibited by an unstirred layer of solution around the cell can become much greater than that of the membrane itself. If the unstirred layer is of considerable thickness, this problem can render the membrane permeability for a gas undetectable.

Unstirred layer thicknesses previously observed vary between 0.5 μm and several hundreds of microns, so it is often unknown what the thickness of this layer is for a given preparation under a given condition. The present paper reports a new principle allowing one (1) to determine the thickness of unstirred layers, (2) to define the effect of the unstirred layers on transport measurements, and (3) to quantitatively correct measured permeabilities for this effect. We have recently introduced a new technique (Wunder et al. 1997; Wunder & Gros, 1998; Forster et al. 1998; Endeward & Gros, 2005; Endeward et al. 2006b, 2008), which allows rather precisely the determination of the permeability of red blood cells, or isolated cells in suspension, or cell layers, for CO2, which is a gas for which some membranes exhibit an especially high permeability and which accordingly is especially difficult to assess. The principle of this method is to observe the exchange of 18O between CO2, HCO3 and water via a special inlet system to a mass spectrometer. This allows one to observe a slow decay of the species C18O16O in solution, which extends over half an hour or more and whose kinetics can be used to determine the permeabilities of the cell membrane to CO2 as well as to HCO3. A major advantage of this technique is that the measured signal is extremely slow, e.g. in comparison to that of a stopped flow experiment. In the latter, the kinetics to be observed occurs within a few tens of milliseconds, making the dead time of the instrument critical. Both techniques, on the other hand, are susceptible to unstirred layers around the cells studied. The novel principle of estimating unstirred layer thickness introduced here is being applied to the determination of CO2 permeability of human red blood cell membranes at 37°C using this 18O exchange technique. We measure CO2 permeabilities of red cells suspended in well-stirred solutions of various concentrations of 60 kDa dextran, thereby varying the viscosity of the solution over a wide range. It is well known that viscosity is an important determinant of unstirred layer thickness (e.g. Flourie et al. 1984; Eisenhans et al. 1984; Berry & Verkman, 1988), and theoretical hydrodynamics predicts that the unstirred layer disappears when viscosity becomes zero (Landau & Lifschitz, 1991; Guyon et al. 2001) while stirring continues. On the basis of this rationale, we plot the measured CO2 permeabilities versus solution viscosity, and extrapolate the curve to zero viscosity to obtain a value of the CO2 permeability that is free of any unstirred layer effect. Of course, this technique can be applied to all other kinds of transport measurements to evaluate the effect of unstirred layers. The present results show that the red cell CO2 permeabilities previously reported by us are no more than 30% lower than the ones obtained here after this correction, and thus are of the correct order of magnitude. The experimental data furthermore show that a proportional relation exists between viscosity and unstirred layer thickness, which is in agreement with hydrodynamic considerations.

The experimental data are complemented by model calculations, in which the situation of 18O exchange between bulk solution, unstirred layer, red cell membrane and red cell interior is simulated. The results essentially confirm the results obtained by extrapolation of the experimental data, but show that the effect of the diffusion resistance exerted by the extracellular unstirred layer is augmented by the intracellular unstirred layer resulting from the lack of intracellular mixing, which enhances the total diffusion resistance to be overcome by CO2 in addition to the membrane resistance. Quantitation of the apparent intracellular unstirred layer thickness from this model is another novel result presented here.

The results are of significance for the ongoing discussion about the existence of gas channels in red cell and other cell membranes, as well as about the value of the CO2 permeability in membranes that lack gas channels. Therefore, these topics are discussed in the light of the present results.


  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Appendix

CO2 permeability by 18O exchange

The principle of this technique, which is an extension of the method developed by Itada & Forster (1977), has previously been described in detail (Wunder et al. 1997; Endeward & Gros, 2005). The entire measurement occurs at complete chemical equilibrium and constant pH, intra- as well as extracellularly. Briefly, chemical, but not isotopic, equilibrium is established in a solution of NaHC18O16O2, in which the labelled species C18O16O, HC18O16O2 and H218O exist. Due to a slow loss of 18O from the CO2–HCO3 pool into the H2O pool that occurs via the CO2 hydration–dehydration reaction, an extremely slow decay of C18O16O occurs and almost all 18O eventually ends up in the water pool. The kinetics of this decay is greatly accelerated when carbonic anhydrase-containing cells such as red blood cells are added, because labelled CO2 (predominantly in a first faster phase) and HCO3 (predominantly in a second slower phase) enter the cells, where due to the intracellular carbonic anhydrase, CA, the hydration–dehydration reaction is much faster and consequently a much more rapid loss of 18O from CO2 into the water pool occurs. The time course of C18O16O is followed using the reaction chamber shown in Fig. 1. This chamber (volume 1.7 ml) is filled with an isotonic solution containing 25 mm HCO3 labelled at 1% with 18O. The solution is adjusted to pH 7.4, kept at the desired temperature and is stirred by a stirring bar driven by an external magnetic stirrer at maximal speed (1 100 rpm). The concentration of C18O16O in this solution is continuously monitored by the mass spectrometer connected to the chamber via an inlet system consisting of a 25-μm-thick teflon membrane supported by a sintered glass disc. The very small fraction of the C18O16O in the solution diffusing into the high vacuum of the mass spectrometer gives a continuous signal indicating the (extracellular) concentration of C18O16O in the solution. After an initial phase of the decay of C18O16O in cell-free solution (first part of the original record of Fig. 2A), red cells are added, which leads to the mentioned biphasic acceleration of C18O16O decay (second and third parts of the record of Fig. 2A). This type of record is used to calculate inline image and inline image by numerically solving the differential equations that describe the entire process in combination with a fitting procedure (Wunder et al. 1997; Endeward & Gros, 2005). Below, we call this entire program the ‘evaluation program’. It may be noted that even with red cells present in the reaction chamber (0.03% v/v) but in the absence of CA in the extracellular fluid, the decay of C18O16O is still slow, taking >30 min to reach isotopic equilibrium. This makes it easily possible to follow the process by the mass spectrometer, which in conjunction with the inlet system, has a response time of 3 s (Endeward & Gros, 2005). This minor delay is corrected for in the calculations by a convolution. Each inline image determination was repeated at least 10 times with a new sample of RBC.


Figure 1. Reaction chamber connected to the mass spectrometer There is continuous diffusion of very small amounts of C18O16O and C16O2 from the (extracellular) solution in the reaction chamber across the 25 μm thick Teflon membrane into the high vacuum of the mass spectrometer, the rate of which is proportional to the concentration of both species in the fluid. Thus, the concentration of both gases in the solution is monitored continuously. Before red cells are added to the solution, pH is adjusted to 7.40. Chamber volume is 1.70 ml.

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Figure 2. Time course of decay of C18O16O in RBC suspension A, original mass spectrometric recording of an experiment with red cells (black dashed line), which were added at about 100 s, giving a characteristic biphasic acceleration of C18O16O decay. Temperature 37°C. Superimposing the experimental curve is the theoretical curve obtained from the fitting procedure of the evaluation program (grey dashed line). B, the black dashed curve represents the result of a mathematical simulation of 18O exchange under the geometrical conditions indicated in Fig. 3. This curve was then subjected to the same evaluation program with fitting procedure as the curve in A. The grey dashed curve obtained from this superimposes the simulated curve, indicating excellent agreement of the two curves. The fitting procedure of the evaluation program not only yields the grey curves shown, but also the values of inline image and inline image that fit the black curves best.

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Viscosity of dextran solutions

An Ubbelohde-type viscometer (Schott-Geräte, 55014 Mainz, Germany) was used to determine the viscosities of water, 125 mm saline and solutions of dextran in saline. Temperature was adjusted by placing the viscometer into a temperature-controlled water bath. Each determination was repeated five times.

Human red blood cells

Human red blood cells were taken freshly from members of the laboratory after their informed consent, washed three times in physiological saline and controlled for haemolysis before being used in the mass spectrometric experiments. In the final suspension in the measuring chamber (Fig. 1) cells were present at a hematocrit of ∼0.03%. To ensure complete inhibition of any extracellular CA arising from minor haemolysis that may occur during the mass spectrometric experiment, all measurements were conducted in the presence of the extracellular CA inhibitor STAPTPP (1-[5-sulfamoyl-1,3,4-thiadiazol-2-yl-(aminosulfonyl- 4-phenyl)]-2,4,6-trimethyl-pyridinium perchlorate) (Casey et al. 2004; found in mass-spectrometric experiments to be membrane-impermeable during exposure of RBC for at least up to 1 h), at a final concentration in the chamber of 1 × 10−5m.


Standard mass spectrometric measurements were performed in 125 mm NaCl to which NaHCO3 labelled at 1% with 18O was added to give a total bicarbonate concentration of 25 mm and isotonicity. Viscosity measurements were performed in 125 mm NaCl solutions with dextran concentrations between 0 and 10 g% (w/v). Dextran was dextran FP 60 research grade with a mean molecular mass of 60 kDa and a molecular mass distribution between 54–66 kDa (Serva Electrophoresis GmbH, Carl-Benz-Str. 7, 69115 Heidelberg, Germany).

The mathematical model

The purpose of the model is to simulate the molecular events underlying the 18O exchange process in the geometrical situation of a red cell with cell interior and cell membrane, an unstirred layer of defined thickness upon the membrane, and a well-stirred surrounding bulk solution of finite volume. These compartments are illustrated in Fig. 3 together with the processes that are considered in each compartment: diffusion in the RBC interior, across the RBC membrane, and in the unstirred layer, while diffusion is neglected in the bulk solution. Reaction between the molecular species involved C18O16O, HC18O16O2, H218O and their unlabelled counterparts occurs in all three fluid compartments. While the reaction equations take into account both labelled and unlabelled species, transport processes consider the labelled species only; this is permissible because there is complete chemical equilibrium in the entire system, and thus any movement of a labelled molecule in one direction is accompanied by movement of the unlabelled molecule in the opposite direction. For the same reason no diffusion potentials will arise and no movements of H+ will occur. The degree of labelling of CO2 and HCO3 is 0.01 when the NaHC18O16O2 is dissolved, but thereafter decreases continuously. The 18O labelling of the huge water pool (55 moles/L (M)) after isotopic equilibrium has been reached of course is magnitudes smaller.


Figure 3. Model simulation of 18O exchange between a well-stirred bulk solution and human red blood cells suspended in it The bulk solution is separated from the red cell membrane by an extracellular unstirred layer, USLe, of a thickness that was varied between 0 and 1 μm. The length of the red cell compartment, i.e. the diffusion path within the red cell, was taken to be the half-thickness of the red cell, 0.8 μm, as introduced for purposes of diffusion calculations by Forster (1964). The volume ratio of red cells to extracellular solution was ∼0.0003 : 1. The arrows indicate diffusion of the labelled species C18O16O, HC18O16O2 and H218O between the volume elements into which each compartment was divided (usually 20–50) and across the red cell membrane. ‘Reaction’ indicates the reaction processes described by eqns (3)(5). While the same reactions occur in the bulk solution, diffusion is not considered there as this compartment is well stirred. Deviations from the situation depicted in the figure are (i) the RBC interior is alternatively considered well-stirred, no diffusion occurs then within the RBC, and USLi= 0, (ii) the extracellular unstirred layer is omitted, setting USLe= 0. The results for these 3 situations are given in Table 1. Constants used in the model calculation are: half-thickness of the red cell 0.8 μm, RBC volume to bulk solution volume ratio 0.0003, inline image in saline 2.4 × 10−5 cm2 s−1, inline image inside RBC 1.2 × 10−5 cm2 s−1 (Gros & Moll, 1971), inline image in saline 1.2 × 10−5 cm2 s−1 (Gros et al. 1976), inline image inside RBC 0.6 × 10−5 cm2 s−1 (same reduction by intracellular Hb assumed as in the case of inline image), pK1' 6.10, pHe 7.40, pHi 7.20, ku 0.15 s−1, ACA inside RBC 20 000.

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Diffusion processes are in all cases described by:

  • image(1)

where X is the molecular species considered, [X] its concentration (m), m is the amount of X (moles), t is time (s), DX the diffusion coefficient of species X (cm2 s−1), A the diffusion area (cm2), x the diffusion path (cm), and PX the permeability for X in case of diffusion across the membrane (cm s−1; for the special treatment of HCO3 transfer across the RBC membrane see Endeward & Gros, 2005).

The 18O reaction that occurs in all three compartments is (Itada & Forster, 1977; Endeward & Gros, 2005):

  • image(2)

The rates of these reaction processes are described by three equations (for details see Endeward & Gros, 2005):

  • image(3)

where ku is the CO2 hydration reaction velocity constant (s−1), ACA is carbonic anhydrase activity in terms of the factor by which CO2 hydration rate is accelerated by CA – 1 (= dimensionless), K1 is the first apparent dissociation constant of carbonic acid (m), and [H+] is the proton concentration.

  • image(4)


  • image(5)

These reaction equations are applied separately to the compartments intra-erythrocytic space, unstirred layer and bulk solution. The contribution of eqn (1) to a change in concentration of any of these species is calculated by dividing the amount of X transferred into the volume element considered by its volume.

A finite-difference approximation technique, the Schmidt method (Crank, 1964, pp. 187–189), was used to solve the model described by eqns (1)–(5) and by Fig. 3. Both compartments, the intra-erythrocytic and the unstirred layer, were divided into 20–50 segments, while the stirred bulk compartment was treated as one large volume unit. All the above differential equations were used in the form of difference equations. Starting conditions were a defined [C18O16O] in the bulk compartment and [C18O16O]= 0 in the unstirred layer and cell interior. At suitable time intervals small enough (<1 μs) to avoid distortions of the calculated time course of C18O16O, the fluxes of the labelled species between the segments and the chemical reactions within each segment were calculated in the first time interval for each consecutive segment. At this – and all subsequent – points of time considered, the quantity of interest was the concentration of C18O16O in the bulk solution, which is the compartment whose [C18O16O] is followed by the mass spectrometer. This calculation was then repeated for subsequent time intervals until the calculations had been repeated for a total time period of at least 180 s. Handing over reaction and diffusion processes and the molecular concentrations across compartment boundaries was accomplished as described by Crank (1964; p. 204). For clarity, we call this program ‘simulation program’.

The quantity obtained, when such a cycle of calculations was completed, was the time course of [C18O16O] over a period of ≥ 180 s. This curve (Fig. 2B), equivalent to an experimental curve of a mass spectrometric measurement (Fig. 2A), was then subjected to the same procedure as used to evaluate inline image and inline image from experimental records (Endeward & Gros, 2005). The necessary parameters ku, K1, ACA, cell-to-chamber volume ratio v, and cell surface-to-volume ratio a, were inserted identically into the simulation and into the evaluation program. Black and grey curves in Fig. 2 show that excellent agreement is achieved by the evaluation program (that includes the fitting procedure) with the experimental curves on one hand (Fig. 2A) as well as with the simulated theoretical curves on the other hand (Fig. 2B). The (true) inline image inserted into the simulation program to produce a curve of C18O16O decay was then compared to the value of the (apparent) inline image obtained by evaluation of the simulated curve of C18O16O decay. Since, in the simulation, unstirred layers are considered, but are neglected in the evaluation program, a decrease in apparent inline image compared to the true inline image is indicative of an effect of the unstirred layers.


  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Appendix

inline image and inline image of human red cell suspensions at various concentrations of dextran

To vary the viscosity of the medium in which fresh normal human red blood cells (RBC) were suspended, dextran concentrations in physiological saline were varied between 0 and 10 g% (w/v). Figure 4 shows the membrane permeabilities for HCO3 and CO2 (inline image and inline image) at 37°C, derived from mass spectrometric measurements as described previously (Wunder et al. 1997; Endeward et al. 2006b). Figure 4A shows that inline image of the RBC membrane in normal saline has a value of ∼1.8 × 10−3 cm s−1, which is in the range of figures reported previously for human RBC at this temperature (Endeward et al. 2008). Importantly, this figure is not dependent on the concentration of dextran, indicating that it is independent of variations in suspension viscosity. In contrast to this, dextran has a profound effect on CO2 permeability, lowering inline image at 10 g% dextran to 1/3 of the control value in the absence of dextran. The control inline image in Fig. 4B amounts to 0.12 cm s−1, which is in good agreement with values previously reported by us for RBC (Endeward et al. 2006b, 2008). It appears therefore that increasing solution viscosity markedly reduces inline image while leaving inline image unaffected, as might be expected on the basis of the much higher membrane permeability for CO2 than for HCO3.


Figure 4. Permeabilities of the human red cell membrane at 37°C, at various concentrations of dextran established in the solution in which the red cells were suspended A shows that inline image does not vary with dextran concentration, but B shows that inline image decreases markedly with increasing dextran concentration.

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Kinematic viscosity ν of solutions of varying dextran concentration

Figure 5 shows viscosity measurements with the Ubbelohde viscometer at 37°C in saline solutions without and with various dextran concentrations up to 10 g%. The viscosity numbers represent kinematic viscosities ν, i.e. are not corrected for solution densities. As expected, viscosity increases drastically with increasing dextran concentration, from a value of ∼0.65 × 10−6 m2 s−1 in saline (tabulated value in pure water 0.70 × 10−6 m2 s−1) up to almost 4 × 10−6 m2 s−1 at 10 g% dextran, i.e. an almost 6-fold increase of ν.


Figure 5. Kinematic viscosities, ν, of NaCl solutions of various dextran concentrations Measured at 37°C in an Ubbelohde viscometer.

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inline image and inline image as a function of viscosity

Figure 6 illustrates how the two permeabilities depend on the viscosity ν of the solutions in which the RBC were suspended. As predicted from Fig. 4, it is apparent that inline image does not vary with viscosity (Fig. 6A), whereas inline image falls to 1/3 of control when viscosity increases 6-fold (Fig. 6B).


Figure 6. Red cell membrane permeabilities at 37°C and at various viscosities of the suspending medium Data points from Figs 4 and 5. A, bicarbonate permeability; B, CO2 permeability. At a viscosity of 4 × 10−6 m2 s−1inline image fell to almost 1/3 of its value in dextran-free saline.

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Results of model calculations

All calculations of Table 1 were performed by inserting a inline image of 0.20 cm s−1 as true CO2 permeability of the RBC membrane. The following major conclusions can be drawn from the table:

Table 1.  Results of model calculations of exchange of 18O between bulk solution, extracellular unstirred layer (USLe) and red cell as shown in Fig. 3
Condition USLiUSLe (μm)PCO2,app from model (cm s−1)PCO2,app from eqn (6) (cm s−1)
  1. The interior of the red cell is either considered perfectly mixed (USLi= 0), or it is considered as a space in which 18O-labelled molecules move by diffusion (USLi+). All calculations are based on a true membrane permeability inline image cm s−1. The apparent inline image values obtained from the model are given in the 5th column. The 6th column gives inline image values estimated on the basis of eqn (6) from the corresponding inline image with USLe= 0 (in parentheses) and the thickness of the USLe, δ, indicated in the 4th column.

0Ideal situation: perfect mixing within RBC, no extracellular unstirred layer00 0.20 
1No intracellular mixing+0 0.15(0.15) 
2No intracellular mixing+
3No intracellular mixing+1.0 0.0960.092
4Perfect intracellular mixing00.50.16 
2No intracellular mixing+0.50.12 
  • 1
    Effects of extracellular unstirred layers. Conditions 1–3 in Table 1 illustrate the effects of the extracellular unstirred layer, USLe, when its thickness δ increases from 0 to 1 μm. As expected, the effective inline image obtained from the model calculation (5th column) decreases with increasing δ. The ‘true’ permeability of 0.20 cm s−1 was chosen because it generates for condition 2 with δ= 0.5 μm (which we show in the discussion below to represent the experimental situation of RBC suspended in saline) the inline image of 0.12 cm s−1 that is actually measured in saline (Fig. 4B). All calculations for conditions 1–3 were performed for the absence of intracellular mixing, i.e. with a normal intracellular unstirred layer, USLi, implying that CO2 moves inside the red cell by diffusion only as indicated in Fig. 3. The rightmost column of Table 1 gives the apparent inline image values as they are derived from eqn (6), when 0.15 cm s−1 is used as the reference value; this approach simply considers the unstirred layer as a diffusion resistance to CO2 and ignores any possible effects of the chemical reaction in the USLe, a process that is implemented in the model simulation. It is apparent that the inline image values for δ= 0.5 μm and 1.0 μm are almost identical between columns 5 and 6.
  • 2
    Effects of intracellular unstirred layers. The effect that the (realistic) absence of intracellular mixing, in comparison to an assumed perfect intracellular mixing, has on the measurement of inline image, is seen by comparing conditions 4 and 2 at the bottom of Table 1. When an extracellular unstirred layer of 0.5 μm is assumed, removal of intra-erythrocytic mixing reduces the apparent inline image from 0.16 to 0.12 cm s−1, i.e. by ∼25%. Comparison of conditions 0 and 1 shows that this effect amounts also to 25% when USLe is taken to be zero.

In conclusion, the model simulations confirm a moderate role of the extracellular unstirred layer as a diffusion resistance to CO2 in addition of that of the membrane itself. In addition, they provide for the first time an estimate of the – also moderate – effect of the fact that gas uptake measurements of red cells necessarily occur with an unstirred cell interior.


  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Appendix

Effect of unstirred layers on red cell membrane permeabilities

We have previously discussed extensively the possible influence of unstirred layers on and around cells upon the apparent membrane permeability for CO2 (Wunder & Gros, 1998; Endeward & Gros, 2005). In principle, to the resistance offered by the membrane itself, a second diffusion resistance due to the layer of unstirred solution on the membrane is added, giving an apparent total resistance that exceeds the mere membrane resistance. In the case of CO2 permeability, this can be expressed as follows:

  • image(6)

where inline image is the apparent permeability of the membrane for CO2, inline image is the permeability obtained when the effect of an unstirred layer is corrected for, δ is the thickness of the unstirred layer, and inline image is the CO2 diffusion coefficient in the solution that surrounds the cells studied. For example, with a inline image of 0.12 cm s−1 (Fig. 4B), a inline image in water of 2.4 × 10−5 cm2 s−1 (Gros & Moll, 1971) and an assumed unstirred layer thickness of 1 μm, eq. 6 gives a inline image of 0.24 cm s−1, i.e. the experimental value would be an underestimate of inline image by a factor of 2. An analogous calculation for HCO3 with inline image of 1.8 × 10−3 cm2 s−1 (Fig. 4A), a inline image in water of 1.2 × 10−5 cm2 s−1 (Gros et al. 1976; corrected for 37°C) and the same unstirred layer thickness shows that, due to the much lower HCO3 permeability compared to that of CO2, the unstirred layer has a negligible effect on this permeability measurement, which is in excellent agreement with the experimental results of Figs 4A and 6A.

Linear experimental relation between the reciprocal of the apparent CO2 permeability and viscosity

Figure 7A gives a plot of the reciprocal of CO2 permeability versus kinematic viscosity using the data of Fig. 6B. Figure 7B gives, for comparison, the analogous plot of data (not shown) obtained at 23°C.


Figure 7. Reciprocal of RBC CO2 permeability as a function of solution viscosity ν A, 37°C. Regression equation: inline image (s.d.± 0.43) + 3.48 (s.d.± 0.21) ·ν. r= 0.994. The y-axis intercept corresponds to a inline image of 0.16 cm s−1. Data from Fig. 6B. B, 23°C. Regression equation: inline image (s.d.± 0.58) + 3.84 (s.d.± 0.20) ·ν. r= 0.996. The y-intercept gives a inline image of 0.14 cm s−1.

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It is apparent that for both sets of data, a linear relationship between inline image and viscosity is obtained. At 37°C, the linear regression yields an intercept on the y-axis of 6.09 ± 0.43 s cm−1. For the data at 23°C, the y-axis intercept is 6.99 ± 0.58 s cm−1. Both correlations are highly significant with P < 0.004.

As mentioned above, the unstirred layer, i.e. a laminar boundary layer, in which the tangential flow velocities decrease drastically along an axis normal to the surface of the cell or object considered, is caused by the viscosity of the fluid and is expected to disappear when the viscosity becomes zero (Landau & Lifschitz, 1991, p. 197). Therefore, extrapolation of the regression lines in Fig. 7 to the intercepts on the y-axes should yield CO2 permeabilities that are corrected for effects of extracellular unstirred layers, inline image. Accordingly, we obtain a value of inline image of 1/6.09 s cm−1, i.e. 0.16 cm s−1, at 37°C. This indicates that in the present mass spectrometric measurements at 37°C inline image is underestimated by ∼30% due to an extracellular unstirred layer effect. This deviation is of minor importance in view of the fact that there have been and are disputes about the order of magnitude of gas and CO2 permeabilities of red cells and other cells and membranes. At 23°C, the intercept of 6.99 s cm−1 in Fig. 7B gives a inline image of 0.14 cm s−1, which is 55% higher than the value of ∼0.09 cm s−1 observed in saline. We conclude that extracellular unstirred layers play a minor role in the measurements of inline image of RBC with the mass spectrometric 18O method.

Unstirred layer thickness of RBC in mass spectrometric experiments and proportional relation between unstirred layer thickness and solution viscosity

Inspection of eqn (6) shows that inline imagevs unstirred layer thickness δ exhibits the same type of function, namely a linear relation with a finite intercept on the inline image axis, as it is seen for inline imagevs viscosity in Fig. 7. This raises the possibility that δ and viscosity are proportional to each other and that this fact underlies the identical types of these functions. It has already been stated that δ vanishes when viscosity becomes zero. We will now explore the possibility of a proportional relation between δ and viscosity both on the basis of the present experimental data and from theoretical considerations.

The data of Fig. 7 together with eqn (6) allow us to calculate the thickness of the unstirred layers covering the red cells in the present measurements in pure saline with the 18O technique. We take from the y-axis intercepts of Fig. 7 the values of inline image as just explained, and together with the values of inline image as obtained experimentally in saline, we use eqn (6) in the form

  • image(7)

to obtain for 37°C: δ= (8.3 − 6.1 s cm−1) × 2.4 × 10−5 cm2 s−1 or δ= 5.3 × 10−5 cm = 0.53 μm.

Application of eqn (7) to the data at 23°C yields an unstirred layer thickness at this temperature of 0.70 μm, when a CO2 diffusion coefficient of 1.7 × 10−5 cm2 s−1 is used for 23°C (Gros & Moll, 1971). Rather similar values of RBC unstirred layer thickness between 0.7 and 0.9 μm have been reported by Holland et al. (1985) from stopped flow measurements of O2 uptake by RBC.

By applying eqn (7) to all data points at various viscosities (i.e. dextran concentrations) of Fig. 7, one can plot δversusν as shown in Fig. 8A and B. It is seen that indeed a very satisfactory proportionality exists between unstirred layer thickness and kinematic viscosity. At both temperatures the intercept on the y-axis is not significantly different from zero (P > 0.4), while both regression coefficients are significantly different from zero with P < 0.004. This confirms that, by varying solution viscosity by suitable concentrations of dextran, we have varied unstirred layer thickness in a proportional manner. This is corroborated by the above observation that the relative increase in unstirred layer thickness from 0.5 μm at 37°C to 0.7 μm at 23°C is roughly identical to the relative increase in water viscosity from ∼ 0.70 × 10−6 m2 s−1 at 37°C to ∼0.94 × 10−6 m2 s−1 at 23°C.


Figure 8. Unstirred layer thickness δ as a function of solution viscosity δ was calculated from the inline image measured at each viscosity and the inline image at zero viscosity obtained from the y-axis intercepts in Fig. 7 using eqn (7). A, 37°C. Regression equation: δ= 0.0001 (s.d.± 0.051) + 0.835 (s.d.± 0.051) ·ν. r= 0.994. B, 23°C. Regression equation: δ= 0.152 (s.d.± 0.098) + 0.652 (s.d.± 0.034) ·ν. r= 0.996. At both temperatures, the y-axis intercept is not significantly different from zero, indicating that a direct proportionality exists between δ and ν.

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In the following we will show that a prediction of proportionality between δ and ν can also be derived from the hydrodynamic treatment of laminar boundary layers (Landau & Lifschitz, 1991; Guyon et al. 2001). For large Reynolds numbers in the bulk flow above cells or other objects, but below the threshold of turbulence, the following relation for the thickness δ of the laminar boundary layer

  • image(8)

has been derived (Landau & Lifschitz, 1991, p. 199; Guyon et al. 2001, p. 386), where is the ‘characteristic length’ of the object considered, e.g. in our case the diameter of the red cell, and Re is the Reynolds number:

  • image(9)

where V is the difference of the velocities of the bulk flow of the solution and the velocity of the red cell or other object under consideration. Inserting eqn (9) into eqn (8) gives:

  • image(10)

In the case of fairly high Reynolds numbers below the threshold of turbulence, V, considered to be the bulk flow observed from a coordinate system originating in the cell under consideration, may be assumed to be proportional to the pressure gradient and inversely proportional to the viscosity η of the fluid:

  • image(11)

where f(p) is a function of the pressure in the bulk solution and ρ is the density of the solution. This corresponds to the treatment underlying the derivation of Hagen–Poiseuille's law. Equation (11) predicts that, when a constant force is exerted by the stirring, V changes approximately inversely proportional to the viscosity of the solution. Alternatively, Landau & Lifschitz (1991; p. 75) present a treatment of this problem for very low Reynolds numbers, which also results in the relation of eqn (11). Thus, inserting eqn (11) into eqn (10), we obtain

  • image(12)

Both treatments just described therefore lead to the prediction that the thickness of the unstirred layer δ is proportional to the kinematic viscosity ν. The other factor in eqn (12) that depends on the properties of the solution, inline image, deviates from the value in water by < 2% even with the highest dextran concentration used, and can thus be considered constant. We conclude that the experimental observation of proportionality between δ and ν is in agreement with theoretical hydrodynamic considerations. This supports the principle used in this study, namely to take measurements of inline image at various solution viscosities and then obtain both its effect on inline image and the thickness of the unstirred layer by extrapolating the regression line to zero viscosity. It should be noted that the δ of eqn (12) is not constant over the entire length of the object considered, but represents rather a maximal value (Landau & Lifschitz, 1991). Therefore, all considerations that follow below should be regarded as rough estimates.

Unstirred layers in suspensions of single cells or vesicles

A further important implication of eqn (12) is that, if all other parameters are kept constant, the following proportionality holds:

  • image(13)

This indicates that the maximal thickness of the unstirred layer increases with the square root of the length of the cell or layer considered. With the diameter of the human red cell of ∼7.5 μm and the unstirred layer thickness of ∼0.5 μm under the present conditions of a very high rate of conventional stirring, we can use eqn (13) to make rough predictions of δ for other preparations used in transport studies. As a wide variety of preparations has been used in measurements of gas transport, we will in the following discuss these results in the light of possible roles of unstirred layers. It should be emphasized that the present figure of 0.5 μm for the unstirred layer thickness of red cells at 37°C crucially depends on the stirring rate and efficiency, which is relatively high in the studies with RBC from our laboratory. A similar efficiency apparently can be obtained in stopped-flow experiments, but, as discussed in the following section, the stirring efficiency appears to be much less in some experiments with larger layers of artificial membranes or of confluent cell cultures.

Vesicle preparations CO2 permeability measurements with membrane vesicles have been reported by Prasad et al. (1998) and by Yang et al. (2000). Vesicles had diameters of 263 nm and between 234 and 302 nm, respectively. In both studies, a stopped-flow apparatus was used to measure the uptake of CO2 by the vesicles by an intravesicular fluorescent pH indicator and, to eliminate a limitation by intravesicular CO2 hydration kinetics, vesicles were preloaded with carbonic anhydrase. Prasad et al. in Zeidel's lab report inline image values of between 0.5 and 1.7 cm s−1 (these are the numbers given in the text, and we have confirmed them by recalculating inline image using the authors' original stopped-flow records and other necessary data as given in their paper, while the values between 0.5–1.7 × 10−3 cm s−1 given in their Table 1 obviously represent misprints). Thus, their range of numbers is roughly of the order of the inline image value of 0.35 cm s−1 reported by Gutknecht et al. (1977) for planar lipid bilayers in a study in which they attempted to circumvent the problem of unstirred layers by adding carbonic anhydrase and bicarbonate to facilitate CO2 diffusion. In contrast, Yang et al. (2000) report a inline image of ∼1 × 10−3 cm s−1, which is three orders of magnitude lower than Prasad's figures.

On the basis of the unstirred layer thicknesses of RBC as reported above and of eqn (13), one arrives, for 23°C, at a rough prediction of an unstirred layer thickness of vesicles of 250 nm in diameter of:

  • image

Such an unstirred layer would present a diffusion resistance to CO2 of inline image of 0.13 × 10−4/1.7 × 10−5 s cm−1, giving per se an apparent inline image of 1.3 cm s−1. In the case of Yang's results this would indicate that the resistance of the membrane to CO2 diffusion is much greater than the resistance exerted by the unstirred layer; in the case of Prasad's result, this would indicate that the unstirred layer affects their result to a minor extent. Gutknecht's studies would favour CO2 permeabilities of (artificial) lipid membranes of around 0.5 cm s−1 but not of 0.001 cm s−1. We note that a problem quite often observed in commercial stopped-flow apparatuses is poor mixing due to an unfavourable geometry of the mixing chamber, which has the effect of generating huge apparent unstirred layers. This may be an explanation of the low inline image observed by Yang et al. (2000).

Red blood cellsYang et al. (2000) present also stopped-flow measurements of inline image of mouse RBC, presumably at room temperature, giving a value of ∼0.012 cm s−1. With the present technique we obtain a inline image of human RBC at 23°C of ∼0.09 cm s−1 and find similar values in mouse RBC (data not shown). Thus, the 18O technique yields a inline image more than 7 times greater than Yang's figure. If this discrepancy were to be explained by an unstirred layer, the red cells in the experiments of Yang et al. (2000) would have to have been covered by an unstirred layer ∼13 μm thick. We propose that the discrepancy might be rather due to insufficient mixing in the stopped-flow apparatus. Ripoche et al. (2006) have measured CO2 uptake by human red cell ghosts using a stopped-flow apparatus, but have not reported permeability figures.

Isolated colonocytes Epithelial cells isolated from proximal and distal guinea pig colon were studied by Endeward & Gros (2005) with the mass spectrometric 18O exchange technique under conditions identical to those used here for RBC. Temperature was 20°C. The spherical colonocytes had a diameter of 10–12 μm, giving, by the type of calculation given above, an unstirred layer thickness of 0.8 μm. This implies a CO2 diffusion resistance exerted by the unstirred layer of 2.9 s cm−1. The reported CO2 permeabilities in proximal colonocytes of 17 × 10−3 cm s−1 and in distal colonocytes of 1 × 10−3 cm s−1 give diffusion resistances of 59 s cm−1 and of 1 000 s cm−1, respectively. In either case, it appears that the expected unstirred layers had little influence on the measured apparent CO2 permeabilities.

Xenopus laevis oocytes The first measurements indicating a role of aquaporin-1 as a CO2 channel have been performed with Xenopus oocytes (Nakhoul et al. 1998; Cooper & Boron, 1998). With a diameter of the oocyte of 1.2 mm, we predict an unstirred layer thickness of the order of 9 μm. This will add a substantial diffusion resistance of the unstirred layer to that of the membrane and tend to apparently decrease any effects of inserting a CO2 channel into the membrane on measured total CO2 diffusion resistance. It may be noted that in oocyte experiments, the rate of stirring may have been less than in the present RBC suspensions because oocytes in our experience are quite susceptible to vigorous mechanical stirring. Thus, the unstirred layer thickness in oocyte experiments might have been greater than this estimate of 9 μm. It is all the more remarkable that clear effects of aquaporin-1 on inline image can be observed.

Unstirred layers on cell layers

We briefly discuss here experiments using extended layers of epithelia to measure membrane CO2 permeability.

Monolayers of intact confluent epithelial cells Studies with Madin–Darby canine kidney (MDCK) cell layers were conducted by Missner et al. (2008). The surface area was 0.33 cm2 indicating a diameter of 3.2 mm. Equation (13) would suggest an unstirred layer of the order of 15 μm, if the stirring was comparable to the one used here, which it probably was not, for the following reason. The authors measured pH profiles in front of the cell layer under conditions of a transcellular net flux of CO2 and estimated the flux from the pH profiles using a complex mathematical model. The unstirred layer thickness is immediately apparent from the pH profiles that extend over ∼150 μm in front of the epithelial layer. An important point here seems that the MDCK cells were grown on a semipermeable support, which again surely was covered by an unstirred layer similar to the 150 μm observed in front of the cell layer. If one takes this unstirred layer on the support, plus the support itself, plus the thickness of the cell layer (neglecting the unstirred layer in front of the cells), one may assume that CO2 permeating this composite layer has to travel across, besides two cell membranes, a layer of water at least 200 μm thick. This water layer exhibits a CO2 diffusion resistance of 1180 s cm−1. A membrane with inline image cm s−1 possesses a resistance of 0.3 s cm−1, one with inline image cm s−1 will have a CO2 diffusion resistance of 10 s cm−1. Even if the CO2 fluxes across these layers were calculated with a precision of 10% from modelling the measured pH gradients, there is clearly no chance that the contribution of the membrane(s) to the total resistance of the composite layer is measurable, no matter what the true membrane permeability for CO2 is.

Intact guinea pig colonic epitheliumEndeward & Gros (2005) reported measurements in which labelled CO2 exchange occurs only through the apical epithelial membrane but not across the entire epithelial layer. They were performed with stripped colonic epithelium slid over a Teflon cylinder about 1 cm in diameter. With this latter value one expects an unstirred layer on the apical side of the epithelium of ∼26 μm, which may be realistic as the stirring speed in these experiments was identically high to that used in the present red cell experiments and the geometry was favourable for achieving good convection on the surface of the epithelium. Endeward & Gros (2005) report for the apical epithelial membrane in these studies a inline image of ∼1 × 10−3 cm s−1 at 20°C. Total apparent CO2 diffusion resistance thus is 1 000 s cm−1, while the resistance due to the unstirred layer of 26 μm would be 153 s cm−1. If this number is realistic, this would indicate that the apical CO2 permeability reported is only 15% lower than the true membrane permeability. However, no direct determinations of unstirred layer thickness have been attempted in this study.

True permeability of the red cell membrane for CO2 derived from model calculations

As discussed above, the extrapolation procedure of Fig. 7 indicates that correction for the extracellular unstirred layer yields a ‘corrected’inline image of 0.16 cm s−1 rather than the experimental value of 0.12 cm s−1. The model calculation confirms this result and predicts a very similar inline image value of 0.15 cm s−1 for the absence of unstirred layers (condition 1). However, the circumstance that the intracellular space is unstirred, which is usually not considered when transmembrane flux measurements are evaluated, causes another error in the experimentally obtained inline image. Table 1 shows that taking into account this fact raises the ‘true’inline image further to a value of 0.20 cm s−1. Thus, lack of correction for extra- and intracellular unstirred layers in the present RBC measurements leads to an underestimation of inline image by 40%. This effect would be much smaller for a substrate with lower membrane permeability such as ions, but even in the case of CO2 transport into red cells, this effect can be considered to be of minor importance. However, in large cells such as Xenopus oocytes, the contribution of USLi will be considerably greater.

The apparent thickness of the intracellular unstirred layer may be estimated by applying analogously eqn (6) to derive δ for USLi. With the intracellular CO2 diffusion coefficient of 1.2 × 10−5 cm2 s−1 (Gros & Moll, 1971) an USLi thickness of 0.2 μm is obtained from comparison of conditions 0 and 1, and of 0.25 μm from comparison of conditions 4 and 2 in Table 1. A value of 0.2 μm represents 1/4 of the maximal intracellular diffusion distance of the half-erythrocyte of 0.8 μm (Forster, 1964). It may be noted that USLi will slightly change when water is taken up or released from the red cell during the cycle the cells pass through between gas exchange in the tissue and in the lung and, therefore, the contribution of USLi to the total diffusion resistance offered by the RBC will vary somewhat.

Basal CO2 permeabilities of red cell and other cell membranes

Early estimates of the CO2 permeability of the RBC membrane by Roughton (1959) and Forster (1969) arrived at numbers of 0.15 cm s−1 and 0.58 cm s−1, respectively, assuming that the CO2 simply and solely moves through the lipid part of the membrane. It may be noted that these numbers are quite consistent with the measurement of Gutknecht et al. (1977) of 0.35 cm s−1 in a lipid bilayer. The results on bilayers are qualitatively confirmed by molecular dynamics simulations of CO2 transport across an artificial membrane, which find that the intrinsic CO2 permeability of the lipid membrane is so high that a gas channel like aquaporin-1 hardly contributes to CO2 permeation (Hub & de Groot, 2006; Wang et al. 2007). It is apparent, however, that many cell membranes do not have such high CO2 permeabilities, when protein CO2 channels are absent or not functional: (1) It has been shown that the apical membranes of the epithelial cells of gastric glands are virtually impermeable to CO2, while the basolateral membranes are highly permeable (Boron et al. 1994; Waisbren et al. 1994); (2) We have shown previously that the apical membranes of surface epithelial cells of the guinea pig colon have a very low CO2 permeability, which coincides with the absence of aquaporin-1 in this membrane (Endeward & Gros, 2005); (3) We have shown that inhibition of both the aquaporin-1 and the Rhesus protein gas channels by 4,4'-diisothiocyanato-2,2'-stilbenedisulphonate (DIDS) drastically reduces RBC inline image, or, even more efficiently, inhibition of the Rhesus channel by DIDS in aquaporin-1 Null RBC reduces inline image of the red cell membrane to the very low value of 0.015 cm s−1 (Endeward et al. 2008). The 18O exchange technique allows us to measure low inline image values with even greater precision than higher inline image's (Endeward et al. 2006b). Unstirred layer problems under these conditions are also less important than in normal RBC as their relative contributions to the total resistance of membrane plus unstirred layer becomes markedly less. Therefore, we believe that there is good evidence that, in the absence of functional CO2 channels, the red cell membrane, as some other cell membranes, has a much lower CO2 permeability than the few artificial lipid bilayers studied so far; (4) HEK cells in suspension have been determined in unpublished experiments in our laboratory (V. Endeward, S. Al-Samir & G. Gros) to have a CO2 permeability of 4 × 10−3 cm s−1, which is 2 orders of magnitude lower than Gutknecht's value of 0.35 cm s−1 for lipid bilayers; (5) Low CO2 permeabilities have also been reported by Blank & Roughton (1960) for artificial monolayers with values ranging from 0.002 to 0.01 cm s−1, and similarly low values have been observed by Ivanov et al. (2004) for O2 permeation through lipid monolayers.

In conclusion, there is good evidence that cell membranes do not in general have high intrinsic CO2 permeabilities due to their lipids, and some, like RBC membranes, achieve a high permeability only through the presence of protein CO2 channels. It is entirely unclear whether the low intrinsic inline image is due to a lipid composition or structure quite different from that of the artificial lipid bilayers that have been studied, or whether this is caused through a blockade of gas transfer by CO2-impermeable proteins within the membrane.

Summary of the evidence for protein CO2 channels in membranes

The first demonstrations that aquaporin-1 may be a channel for CO2 were by Nakhoul et al. (1998) and by Cooper & Boron (1998) and later by Uehlein et al. (2003), who expressed aquaporin in Xenopus oocytes. They studied the kinetics of CO2 uptake into oocytes via measurement of pHi, and found an acceleration in the presence of aquaporin-1 in the membrane. These effects were visible although, as explained above, certainly a significant unstirred layer was present, whose thickness depends on the stirring efficiency in these experiments. The aquaporin-1 effects were even more visible in a later study in which surface pH changes were measured with a pH microelectrode (Endeward et al. 2006b). These experiments again confirmed that CO2 penetrates into oocytes more rapidly when aquaporin-1 is expressed in the membrane.

In vesicles into which aquaporin-1 was reconstituted, Prasad et al. (1998) in Zeidel's laboratory found a significant increase in inline image when aquaporin-1 was present, while Yang et al. (2000), in very similar experiments, did not. However, Prasad et al. (1998) found reasonable inline image control values, while those of Yang et al. (2000) were clearly far too low, which might perhaps be explained by an inadequate mixing in their stopped-flow apparatus. Thus, it appears that there is good evidence from vesicle studies of Prasad et al. (1998) for aquaporin-1 being a CO2 channel.

Our own experiments with red cells conducted by 18O exchange mass spectrometry have recently been criticized as being possibly compromised by an unstirred layer of unknown thickness (Missner et al. 2008), although Blank & Ehmke (2003), using an entirely different technique, provided independent evidence for CO2 channels in RBC membranes. The present work shows that the unstirred layer in our method is 0.5 μm thick and thus in the range of unstirred layer thicknesses of red cells observed in other studies. An extracellular unstirred layer of this magnitude leads to an underestimation of RBC inline image at 37°C by at most 30%, giving a true value in normal saline of 0.16 cm s−1 rather than the experimental 0.12 cm s−1. Values that are reduced by the absence of aquaporin-1 or Rhesus complex (Colton Null or Rh Null blood) as reported by Endeward et al. (2006a) and Endeward et al. (2006b, 2008), will be underestimated to a lesser extent, because the relative contribution of the unstirred layer to the apparent total membrane resistance will be smaller. Therefore, upon correction of the unstirred layer effects, the difference between control inline image values and inline image values of aquaporin-1 Null or Rhesus protein Null red cells will even become somewhat greater than published. We conclude from this that these RBC results, therefore, provide clear evidence for the roles of aquaporin-1 and Rhesus protein as CO2 channels. The molecular dynamics calculations of Hub & de Groot (2006) and Wang et al. (2007) also show clearly that aquaporin-1 constitutes a channel for CO2, the unsolved problem being that in their calculations the model membrane possesses an even higher permeability for CO2 than is mediated by the channels.

A recent study by Missner et al. (2008) from Pohl's and Zeidel's laboratories has attempted to study the question whether aquaporin-1 is a CO2 channel by measuring CO2 flux across confluent layers of MDCK (Madin–Darby canine kidney) cells with or without expression of aquaporin-1. They observed no difference between the presence and absence of aquaporin-1 and concluded that this protein may not be a gas channel. As explained above, the experimental arrangement in these studies was such that any membrane resistance for CO2, be it rather high or rather low, will be negligible compared to the total diffusion resistance of the aqueous layer that has to be crossed by CO2. These data therefore do not appear to be suitable to contribute to the question of gas channels.

Other studies that did not obtain evidence for a role of aquaporin-1 or Rhesus protein in CO2 transfer are the stopped-flow studies with red cells by Yang et al. (2000) and those with red cell ghosts by Ripoche et al. (2006). Aside from potential mixing problems as already discussed, it should be noted that the uptake of CO2 by a red cell is extremely fast, just a few milliseconds if there is no buffer inside the cell, and time required for uptake increases to about 100 ms when the very high buffer capacity of haemoglobin is available (Gros & Moll, 1971; Endeward et al. 2008). This indicates that experiments with ghosts become increasingly more difficult when intraerythrocytic buffer capacity decreases. Therefore, with moderate buffer concentrations, the process of CO2 uptake may be relatively fast and a major initial part of it may escape observation in the stopped-flow apparatus.

Finally it should be mentioned that in the intact lung Yang et al. (2000) and Swenson et al. (2002) could detect neither an effect of aquaporin-1 deficiency nor of aquaporin-1 inhibition on CO2 exchange. This is, however, not surprising in view of our previous calculations that even significant reductions of erythrocyte membrane permeability to CO2 are not expected to seriously hamper gas exchange across the lung (Endeward et al. 2006b, 2008), mainly because a major decrease in permeability may be compensated by a minor increase in veno-arterial CO2 partial pressure difference.


  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Appendix
  • Berry CA & Verkman AS (1988). Osmotic gradient dependence of osmotic water permeability in rabbit proximal convoluted tubule. J Membr Biol 105, 3343.
  • Blank ME & Ehmke H (2003). Aquaporin-1 and HCO3Cl transporter-mediated transport of CO2 across the human erythrocyte membrane. J Physiol 550, 419429.
  • Blank M & Roughton FJW (1960). The permeability of monolayers to carbon dioxide. Trans Far Soc 56, 18321841.
  • Boron WF, Waisbren SJ, Modlin IM & Geibel JP (1994). Unique permeability barrier of the apical surface of parietal and chief cells in isolated perfused gastric glands. J Exp Biol 196, 347360.
  • Casey JR, Morgan PE, Vullo D, Scozzafava A, Mastrolorenzo A & Supuran CT (2004). Carbonic anhydrase inhibitors. Design of selective, membrane-impermeant inhibitors targeting the human the human tumor-associated isozyme IX. J Med Chem 47, 23372347.
  • Cooper GJ & Boron WF (1998). Effect of pCMBS on CO2 permeability of Xenopus oocytes expressing aquaporin 1 or its C189S mutant. Am J Physiol Cell Physiol 275, C1481C1486.
  • Crank J (1964). The Mathematics of Diffusion. Oxford University Press, Oxford .
  • Eisenhans B, Zenker D & Caspary WF (1984). Guaran effect on rat intestinal absorption. A perfusion study. Gastroenterology 86, 645653.
  • Endeward V, Cartron J-P, Ripoche P & Gros G (2006a). Red cell membrane CO2 permeability in normal human blood and in blood deficient in various blood groups, and effect of DIDS. Transfus Clin Biol 13, 123127.
  • Endeward V, Cartron J-P, Ripoche P & Gros G (2008). RhAG protein of the Rhesus complex is a CO2 channel in the human red cell membrane. FASEB J 22, 6473.
  • Endeward V & Gros G (2005). Low carbon dioxide permeability of the apical epithelial membrane of guinea pig colon. J Physiol 567, 253265.
  • Endeward V, Musa-Aziz R, Cooper GJ, Chen L-M, Pelletier MF, Virkki L, Supuran CT, King LS, Boron WF & Gros G (2006b). Evidence that aquaporin 1 is a major pathway for CO2 transport across the human erythrocyte membrane. FASEB J 20, 19741981.
  • Flourie B, Vidon N, Florent CH & Bernier JJ (1984). Effect of pectin on jejunal glucose absorption and unstirred layer thickness in normal man. Gut 25, 936941.
  • Forster RE (1964). Rate of gas uptake by red cells. In Handbook of Physiology, section 3, The Respiratory System, vol. 1, Respiration, ed. FennWO & RahnH. The American Physiological Society, Washington , DC .
  • Forster RE (1969). The rate of CO2 equilibration between red cells and plasma. In CO2: Chemical, Biochemical, and Physiological Aspects, ed. ForsterRE, EdsallJT, OtisAB & RoughtonFJW, pp. 275284. National Technical Information Service. NASA SP-188. Washington , DC .
  • Forster RE, Gros G, Lin L, Ono Y & Wunder M (1998). The effect of 4,4′-diisothiocyanato-stilbene-2,2′-disulfonate on CO2 permeability of the red blood cell membrane. Proc Natl Acad Sci U S A 95, 1581515820.
  • Gros G & Moll W (1971). The diffusion of carbon dioxide in erythrocytes and hemoglobin solutions. Pflügers Arch 324, 249266.
  • Gros G, Moll W, Hoppe H & Gros H (1976). Proton transport by phosphate diffusion – a mechanism of facilitated CO2 transfer. J Gen Physiol 67, 773790.
  • Gutknecht J, Bisson MA & Tosteson FC (1977). Diffusion of carbon dioxide through lipid bilayer membranes – effects of carbonic anhydrase, bicarbonate, and unstirred layers. J Gen Physiol 69, 779794.
  • Guyon E, Hulin J-P, Petit L & Mitescu CD (2001). Physical Hydrodynamics. Oxford University Press Inc., New York .
  • Holland RAB, Shibata H, Scheid P & Piiper J (1985). Kinetics of O2 uptake and release by red cells in stopped-flow apparatus: effects of unstirred layer. Respir Physiol 59, 7191.
  • Hub JS & De Groot BL (2006). Does CO2 permeate through aquaporin-1? Biophys J 91, 842848.
  • Itada N & Forster RE (1977). Carbonic anhydrase activity in intact red blood cells measured with 18O exchange. J Biol Chem 252, 38813890.
  • Ivanov II, Fedorov GE, Gus'kova RA, Ivanov KI & Rubin AB (2004). Permeability of lipid membranes to dioxygen. Biochem Biophys Res Comm 322, 746750.
  • Landau LD & Lifschitz EM (1991). Lehrbuch der Theoretischen Physik. Band VI, Hydrodynamik. 5. Auflage. Akademie-Verlag, Berlin .
  • Missner A, Kügler P, Saparov SM, Sommer K, Mathai JC, Zeidel ML & Pohl P (2008). Carbon dioxide transport through membranes. J Biol Chem 283, 2534025347.
  • Nakhoul NL, Davis BA, Romero MF & Boron WF (1998). Effect of expressing the water channel aquaporin-1 on the CO2 permeability of Xenopus oocytes. Am J Physiol Cell Physiol 274, C543C548.
  • Prasad GVR, Coury LA, Finn F & Zeidel ML (1998). Reconstituted aquaporin 1 water channels transport CO2 across membranes. J Biol Chem 273, 3312333126.
  • Ripoche P, Goossens D, Devuyst O, Gane P, Colin Y, Verkman AS & Cartron J-P (2006). Role of RhAG and AQP1 in NH3 and CO2 gas transport in red cell ghosts: a stopped-flow analysis. Transfus Clin Biol 13, 117122.
  • Roughton FJW (1959). Diffusion and simultaneous reaction velocity in hemoglobin solutions and red cell suspensions. Progr Biophys Biophys Chem 9, 55104.
  • Swenson ER, Deem S, Kerr ME & Bidani A (2002). Inhibition of aquaporin-mediated CO2 diffusion and voltage-gated H+ channels by zinc does not alter rabbit lung CO2 and NO excretion. Clin Sci (Lond) 103, 567575.
  • Uehlein N, Lovisolo C, Siefritz F & Kaldenhoff R (2003). The tobacco aquaporin NtAQP1 is a membrane CO2 pore with physiological functions. Nature 425, 734737.
  • Waisbren SJ, Geibel JP, Modlin IM & Boron WF (1994). Unusual permeability properties of gastric gland cells. Nature 368, 332335.
  • Wang Y, Cohen J, Boron WF, Schulten K & Tajkhorshid E (2007). Exploring gas permeability of cellular membranes and membrane channels with molecular dynamics. J Struct Biol 157, 534544.
  • Wunder M, Böllert P & Gros G (1997). Mathematical modelling of the role of intra- and extracellular activity of carbonic anhydrase and membrane permeabilities of HCO3, H2O, and CO2 in 18O exchange. Isotopes Environ Health Stud 33, 197205.
  • Wunder MA & Gros G (1998). 18O exchange in suspensions of red blood cells: determinations of parameters of mass spectrometer inlet system. Isotopes Environ Health Stud 34, 303310.
  • Yang B, Fukuda N, Van Hoek A, Matthay MA, Ma T & Verkman AS (2000). Carbon dioxide permeability of aquaporin-1 measured in erythrocytes and lung of aquaporin-1 null mice and in reconstituted proteoliposomes. J Biol Chem 275, 26862692.


  1. Top of page
  2. Abstract
  3. Methods
  4. Results
  5. Discussion
  6. References
  7. Appendix


We are indebted to Ms Sisko Bauer and Mr Werner Zingel for skilled technical assistance. We thank Professor Claudiu Supuran for a generous gift of the extracellular carbonic anhydrase inhibitor STAPTPP. We thank the Deutsche Forschungsgemeinschaft for support of this project (Gr 489/19-1-2).