Nitric oxide uptake by erythrocytes is primarily limited by extracellular diffusion not membrane resistance.

The process of NO transfer into erythrocytes (RBCs) is of critical biological importance because it regulates the bioavailability and diffusional distance of endothelial-derived NO. It has been reported that the rate of NO reaction with oxyhemoglobin (Hb) within RBCs is nearly three orders of magnitude slower than that by equal amounts of free oxyhemoglobin. Consistent with early studies on oxygen uptake by RBCs, the process of extracellular diffusion was reported to explain this much lower NO uptake by RBC encapsulated Hb (Liu, X., Miller, M. J., Joshi, M. S., Sadowska-Krowicka, H., Clark, D. A., and Lancaster, J. R., Jr. (1998) J. Biol. Chem. 273, 18709-18713). However, it was subsequently proposed that the RBC membrane provides the main resistance to NO uptake rather than the process of extracellular diffusion (Vaughn, M. W., Huang, K. T., Kuo, L., and Liao, J. C. (2000) J. Biol. Chem. 275, 2342-2348). This conclusion was based on competition experiments that were assumed to be able to determine the rate constant of NO uptake by RBCs without extracellular diffusion limitation. To test the validity of this hypothesis, we theoretically analyzed competition experiments. Here, we show that competition experiments do not eliminate the extracellular diffusion limitation. Simulation of the competition data indicates that the main resistance to NO uptake by RBCs is caused by extracellular diffusion in the unstirred layer surrounding each RBC but not by the RBC membrane. This extracellular diffusion resistance is responsible for preventing interference of NO signaling in the endothelium without the need for special NO uptake by intracellular hemoglobin or a unique membrane resistance mechanism.

Nitric oxide (nitrogen monoxide, NO) has been recognized as a critical physiological mediator in the regulation of vascular tone (1). It is generated in vascular endothelium by a specific arginine-dependent NO synthase, NOS3. Free diffusion has been considered to be the main process that determines how NO travels from the site of its formation to target sites where it exerts its critical physiological functions (2,3).
In the vascular lumen NO can be scavenged by hemoglobin subunits (Hb) 1 within RBCs. The reaction of NO with myoglobin and hemoglobin is rapid (bimolecular rate constant for oxyMb is k Mb ϭ 3 ϫ 10 7 M Ϫ1 s Ϫ1 and for oxyHb is k Hb ϭ 6-9 ϫ 10 7 M Ϫ1 s Ϫ1 ) (4,5). It has been reported that 10 M cell-free Hb is enough to trap almost all NO production generated from endothelial cells and abolish NO-mediated vasodilation (6,7), while blood contains about 8 mM Hb, which is much higher than the quantity of Hb required to completely eliminate NO-mediated functional effects. The rapid reaction between NO and oxyhemoglobin raises the question of how NO can escape from the large quantity of Hb in the blood vessel lumen to exert physiological effects in the blood vessel wall after it is generated from endothelial cells (2,8). This has led to recent efforts to understand and theoretically model the reaction process between NO and Hb (9 -12).
It was observed that the rate of NO reaction with RBCenclosed Hb is nearly three orders of magnitude slower than the rate of NO reaction with free Hb (9). Based on a diffusionlimited reaction model where the limiting step for NO reaction with RBC-enclosed Hb is diffusion of NO from the solution to the surface of RBCs, it was found that the calculated rate constant for the reaction between NO and RBC Hb is very close to that experimentally determined with direct electrochemical measurements. Recently, these experimental results were verified in competition experiments comparing the scavenging of NO by free Hb to that of RBC-enclosed Hb (13); however, it was concluded that the RBC membrane is the main resistance to NO transfer into RBCs. This was based on the following competition Equation 1, which was derived under the assumption that much of extracellular diffusion resistance is eliminated in competition experiments because NO is uniformly generated in extracellular solution, and a high hematocrit (Ͼ5%) was used in the experiments (13,14). Hb] ex is measured experimentally and then the ratio of k RBC / k Hb is determined from Equation 1 using these experimental data. It was found that the ratio of k RBC /k Hb has a large increase as hematocrit changes from 1.6 to 5%, and then the ratio slightly increases as hematocrit changes from 5 to 16%. The large increase of k RBC as hematocrit changes from 1.6 to 5% was attributed to the diminution of extracellular diffusion resistance, and therefore the value of k RBC , measured in the high hematocrit region, was considered to be caused only by RBC membrane resistance.
The controversy about the resistance of RBCs to NO transfer is analogous to that relating to explanations for RBC oxygen uptake. More than 70 years ago, it was discovered that RBC oxygen uptake was considerably slower than that of an equivalent concentration of free Hb (15). Early investigators suggested that oxygen uptake by RBCs is limited mainly by the transport of oxygen across the RBC membrane and intracellular diffusion of oxygen based on the assumption that extracellular oxygen gradients are not present in rapid mixing devices (16,17). However, it is now accepted that effective diffusion layers around RBCs still exist in rapid mixing devices (18,19) and are mainly responsible for the resistance of RBC oxygen transfer (18 -23).
Since the assumption of competition experiments was not based on strict theoretical analysis, one may ask the following two questions. 1) Is NO concentration really uniformly distributed in solution in the competition experiments? 2) Does the value of k RBC at high hematocrit (from 5 to 16%) really represent the rate constant of transmembrane diffusion? To answer the two questions, we have theoretically analyzed competition experiments in this study. NO concentration, NO flux at RBC membrane, the formation rate of extracellular metHb, and the ratio of k RBC /k Hb are calculated and shown at different membrane permeabilities (P m ) and hematocrits (Hct). Our results show that NO concentration is not uniform in solution in competition experiments. Computer simulation of competition data indicates that RBC NO uptake is primarily limited by the extracellular diffusion resistance rather than the RBC membrane resistance.

MATERIALS AND METHODS
Model Assumptions-The disc shape of RBC is replaced with a spherical shape for simplifying the theoretical analysis. The effective radius of the spherical RBC, r 0 , is determined from the surface area of the disc shape (14). In this model, the spherical RBC is fixed at the center of a limited spherical volume or container with a radius r 1 . The radius of the container is defined by the hematocrit as the ratio of the RBC volume to the container volume,r 0 3 / r 1 3 . For the competition experiments, the RBC is surrounded with Hb solution that fills the space between the surface of the RBC and the inner wall of the container (Fig. 1). After adding an NO donor to the extracellular Hb solution, the donor uniformly distributes in the whole extracellular solution. NO released from the donor either reacts with Hb in the extracellular solution to form metHb or transfers into the RBC. NO that enters into the RBC is immediately scavenged by the concentrated intracellular Hb so that NO concentration inside the RBC is maintained very close to zero. Since NO is slowly released from the NO donor and extracellular Hb concentration is much greater than NO concentration in competition experiments, the extracellular Hb can be considered to be homogeneously distributed in the solution. With these assumptions we have analytically derived expressions to predict extracellular NO concentration, NO flux at the RBC membrane, the formation rate of extracellular metHb, and the ratio of k RBC /k Hb .
Mathematical Analysis of Diffusion in Competition Experiments-Based on the above assumptions, the following diffusion-reaction equation and boundary conditions (Equations 2-4) are used to describe the competition experiment for any hematocrit (from 0 to 100%), [Hb] ex is extracellular Hb concentration; and r 0 and r 1 are radii of the RBC and the spherical container as shown in Fig. 1, respectively. In solution, the change of NO concentration at any location between r ϭ r 0 and r ϭ r 1 can be caused by NO diffusion, NO generation, and NO consumption at that location, which is expressed by the first, second, and third terms in Equation 2. At the boundary r ϭ r 0 , the diffusion rate of NO into the RBC surface should be equal to the rate of NO transfer across the RBC membrane, so we have Equation 3. At the boundary r ϭ r 1 , there is no net NO flow into or out from the container surface, so the gradient of NO concentration at r ϭ r 1 should be equal to zero as expressed in Equation 4.
Computer Simulations-All simulated curves in this study were calculated from the analytical expressions derived in the next section on a pentium III personal computer with QBasic programs.

The Effect of Hematocrit on NO Concentration
where [NO] b is the NO concentration in solution in the absence of RBCs, Diagram illustrating the competition model of the NO reaction with free and RBC Hb. A spherical container of radius r 1 is filled with free extracellular Hb solution, and a Hb-containing RBC with radius r 0 is fixed at the center of the container. The radius r 1 is determined by the Hct from the equation Hct ϭ (r 0 /r 1 ) 3 .
according to the definition of hematocrit, Hct ϭ r 0 3 / r 1 3 , we can replace r 1 with Hct in Equation 7. NO concentration distribution profiles around the RBC for different P m and Hct values are shown in Fig. 2, A and B. In the simulations, we used [Hb] ex ϭ 10 M, k Hb ϭ 89 M Ϫ1 s Ϫ1 (20°C) (5), D NO ϭ 2600 m 2 /s (25°C) (24). Since the molar volume of NO is close to the molar volume of oxygen, their diffusion constant would be similar (25). The diffusion constant of oxygen in solution at 25°C is 2420 m 2 /s (26); therefore, it is reasonable to take the value of D NO ϭ 2600 m 2 /s as the NO diffusion constant at 25°C. In competition experiments, it was assumed that when hematocrit was smaller then 5%, the extracellular diffusion resistance becomes significant as hematocrit decreases, but k RBC calculated from the competition equation did not include any extracellular diffusion resistance as the hematocrit was greater than 10% (13). The profiles of NO concentration that were drawn according to the assumption for competition experiments are shown in Fig. 2C. In this assumption, the NO concentration distribution in the diffusion layer must have a large change to eliminate the inhomogeneous NO concentration distribution as hematocrit increases from 0.2 to 5%.
The Effect of Hematocrit on the Flux of NO Transfer into RBCs-In order to further test how hematocrit affects the extracellular diffusion resistance, we derived the analytical expression for the flux of NO transfer into RBCs and analyzed how the flux varies with hematocrit. If the flux largely increases with hematocrit or the extracellular diffusion resistance largely decreases with hematocrit, then membrane resistance may become the limiting step to NO transfer into RBCs.
The flux of NO diffusion into the RBC is given by Equation 11, In competition experiments, NO donor or NO is used as a probe to measure the ratio of k RBC /k Hb . It is assumed that the physical properties of the RBC membrane and solution diffusion resistances are independent of [NO]  dF n /dr 1 ϭ 12͓f 2 P m r 0 r 1 /Y͑f, r 1 , r 0 ͔͒ 2 /͑k Hb ͓Hb͔ ex r 0 D NO ͒ Ͼ 0 (Eq. 13) Since dF n /dr 1 is always larger than zero, F n will increase as r 1 increases and decrease as r 1 decreases. F n will decrease as Hct increases because r 1 is inversely proportional to Hct 1 ⁄3 (Fig.  3). No matter whether the membrane permeability P m is as high as 0.9 m/s or as low as 0.0018 m/s, F n always decreases with the increase of Hct. Since F n does not increase as hematocrit changes from 0 to 5%, extracellular diffusion resistance cannot be appreciably reduced by increasing the hematocrit.
Expressions    Fig. 4. The points in Fig. 4, A-C were read from Fig.  2(a), Fig. 4(a), and Fig. 5(a)    shows that the experimental data are in good agreement with curve a, which was calculated under the assumption of extracellular diffusion limitation (P m ϭ 0.9 m/s). However, curve b calculated under the assumption of membrane resistance limitation (P m ϭ 4.15 ϫ 10 Ϫ4 m/s) is very different from the experimental data. DISCUSSION Our model predicts that the NO concentration at the RBC surface, [NO] m , is close to zero if P m ϭ 0.9 m/s, a value reported by Subczynski et al. (28) for lipid bilayer membrane. In this case, NO uptake by RBCs would be limited by extracellular diffusion (Fig. 2A). At Hct ϭ 0.2%, the distance between the RBC surface to the inner wall of the container, r 1 Ϫ r 0 , is about 23.5 m. There is a layer with a big NO concentration gradient (r Ϫ r 0 Ͻ 6 m) near the RBC, and a region of uniform NO concentration outside the concentration gradient layer between r Ϫ r 0 ϭ 6 m and r Ϫ r 0 ϭ 23.5 m. The radius of the container r 1 decreases with the increase of hematocrit. As Hct ϭ 5%, (r 1 Ϫ r 0 ) is reduced to 5.8 m, and the region of uniform NO concentration disappears. However, the inhomogeneous NO concentration distribution within the concentration gradient layer almost remains unchanged. Thus, in contrast to the assumption of the competition experiments, increasing hematocrit from 0.2 to 5% does not make NO concentration distribution more homogeneous; rather NO concentration distribution is more inhomogeneous. Further increasing hematocrit from 5 to 10% has little effect on the NO concentration gradient near the RBC surface. NO uptake by the RBC is still limited by extracellular diffusion because [NO] m is still close to zero compared with the NO concentration at the inner surface of the container, [NO] c . Fig. 2B shows the profile of NO concentration as the RBC membrane is 500 times lower, or P m ϭ 1.8 ϫ 10 Ϫ3 m/s. Similar to P m ϭ 0.9 m/s, the NO concentration distribution in solution as P m ϭ 1.8 ϫ 10 Ϫ3 m/s is more inhomogeneous, not more homogeneous, as hematocrit increases from 0.2 to 5%. At Hct ϭ 10%, the big NO concentration gradient still exists between the RBC surface and the inner wall of the container. Simply speaking, Fig. 2, A and B show that changing hematocrit from 1.6 to 10% has little effect on extracellular diffusion resistance no matter if P m is high or low. However, the original assumption of competition experiments (Fig. 2C), which predicts a complete conversion of inho-mogeneous NO concentration distribution into homogeneous NO concentration distribution is in conflict with those predicted by the diffusion-reaction equation (Fig. 2, A and B). Therefore, there is no theoretical basis from which to claim that k RBC determined around Hct ϭ 10% does not include any extracellular diffusion resistance.
In fact, there is evidence in the literature for an external diffusional limitation for other small nonelectrolytes, such as O 2 and CO (29,18) across the erythrocyte membrane. Vandegriff and Olson (30) demonstrated that the rate of O 2 uptake by erythrocytes is inversely proportional to the size of RBC (using erythrocytes from various species as well as liposome-entrapped hemoglobin), a result directly predicted by the existence of an unstirred layer (9,30) and not predicted on the basis of an inherent diffusion barrier (9,30). Since NO and O 2 have similar size and solubility, the permeability of RBC membrane to the two small neutral molecules should be similar. In a previous study, we showed that NO uptake by RBCs at a low hematocrit is limited by extracellular diffusion (9). Vaughn et al. (13) also demonstrated that extracellular diffusion is the main resistance to NO uptake by RBCs as Hct ϭ 1.6% in competition experiments. The above theoretical analysis (refer to Fig. 2A) shows that if NO uptake by RBCs is limited by extracellular diffusion resistance at low hematocrit, the extracellular diffusion resistance would have little change by increasing hematocrit from 0.2 to 10%. Thus NO uptake by RBCs in competition experiments of Vaughn et al. (13) would still be limited by extracellular diffusion resistance. Fig. 4 shows that if NO uptake by RBCs is limited by extracellular diffusion, or P m ϭ 0.9 m/s, the calculated curves of [metHb] ex at different sets of hematocrit, [Hb] ex and NO donor concentration are in good agreement with experimental data. As shown in Fig. 5, the experimentally determined ratios of k RBC /k Hb , except the one as Hct ϭ 1.6%, are very close to curve a (P m ϭ 0.9 m/s), indicating that extracellular diffusion is the main resistance to NO uptake by RBCs. The experimentally determined ratio of k RBC /k Hb as Hct ϭ 1.6% largely deviates from curve a. This large deviation is caused by a low initial [Hb] ex (2 M) that was used in the experiment as Hct ϭ 1.6%, because this low [Hb] ex can result in a low ratio of k RBC /k Hb and may cause a larger relative error in an experimental measurement of [metHb] ex . If we let P m ϭ 4.15 ϫ 1 0 Ϫ4 m/s, a very low membrane permeability suggested by Vaughn et al. (14), the simulated curve (curve b) is much lower than the reported ratios of k RBC /k Hb .
The analysis above shows that the rate of NO uptake by RBCs is primarily limited by extracellular NO diffusion in the competition experiments. Mathematically, as long as a nonzero diffusion flux exists around a RBC, the diffusion flux can be always related to a diffusion layer as shown in Equation 19, where D is the diffusion coefficient; c b and c s are bulk concentration and surface concentration, respectively; and ␦ is the thickness of the diffusion layer. Here, the diffusion layer is an imaginary layer, which is used to convert the concentration gradient dc/dr on the RBC membrane into a simple fraction (c b Ϫ c s )/␦. Thus, the thickness of the diffusion layer is always related to the concentration gradient on a given surface. In a competition experiment, NO is generated around each RBC, so the diffusion layer is a layer within the unstirred layer around each RBC. In a blood vessel, the shape of the diffusion layer is much different. The blood flow creates an RBC-free layer near the vessel wall and increased concentration of RBCs near the center of the vessel like a flowing RBC cylinder (7, 10, 11). The endothelium-derived NO needs to diffuse across the RBC-free layer to reach RBCs. In this case, the diffusion layer is a layer within the RBC-free layer surrounding the surface of the flowing RBC cylinder. The diffusion layer can reduce NO uptake by RBCs and enhance the effective NO diffusion distance in the vessel wall (10).
Finally, we point out that numerous previous studies with erythrocytes do not support the possibility that it possesses an inherent diffusion barrier to small nonelectrolytes, certainly not of the magnitude proposed by Vaughn et al. (13,14). The red cell membrane is 40% lipid, which represents a substantial surface of rapid access of NO for diffusion. The lipid bilayer is highly unlikely to possess such a barrier property, since the passive permeability of the RBC to nonelectrolytes is very similar to artificial lipid vesicles (and also other cell types) (31), and the permeability characteristics of lipid vesicles made from whole erythrocyte membrane lipids is quantitatively the same as for synthetic lipids (32). Also, the rate of lateral diffusion of NO in the red cell membrane is similar to phosphatidylcholine vesicles (33). O 2 diffusion through thin films of erythrocytes is similar to that through equivalent thicknesses of free hemoglobin, indicating that the membrane possesses no barrier (34). The presence of the complex cytoskeletal network of the erythrocyte and/or binding of hemoglobin to the cytoskeleton also exerts no influence, since the passive permeability of intact red cells to mannitol and erythritol is the same as inside-out vesicles, which lack this cytoskeleton (35). In addition, if the cytoskeletal network serves as a physical barrier, in order to account for a retardation in entry of a factor of 800 (36) this mechanism would require that such a network would cover all but 1/800 ϭ 0.125% of the total inner surface area of the membrane. The spectrin component is an elongated rod that forms a web with large spaces inbetween (37). As suggested by Huang et al. (36), hemoglobin could bind to the membrane and conceivably cover parts of the surface. However, with 10 6 hemoglobin molecules bound to band 3 per erythrocyte (38) and the size of hemoglobin at 64 ϫ 55 ϫ 50 Å 3 (39), it can be calculated that the maximum surface area covered by the bound hemoglobin will be 45.7 m 2 , whereas the total erythrocyte surface area is 135 m 2 (40). This would not explain a slowing of transmembrane NO movement of nearly 1000-fold.
In conclusion, the process of NO transfer into RBCs is of critical biological importance as it regulates the bioavailability and diffusion distance of endothelial derived NO. Our analysis shows that with increasing hematocrit, the distribution of NO concentration becomes more inhomogeneous rather than more homogeneous. Competition experiments did not eliminate extracellular diffusion resistance in measuring the ratio of k RBC / k Hb under the reported experimental conditions. The rate of NO uptake by RBCs in competition experiments is primarily limited by extracellular diffusion not membrane resistance. The extremely low membrane permeability suggested by Vaughn et al. (14) is not only in conflict with many previous experimental results about O 2 and NO uptake by RBCs, but also inconsistent with their results about the ratio of k RBC /k Hb calculated from the competition equation.