INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

NITRIC OXIDE (NO) is a signal transduction molecule of extreme physiological importance. It is produced in a number of cells through the enzymatic degradation of L-arginine by one of several isoforms of nitric oxide synthase (NOS). NO is a relatively reactive molecule with a short half-life in vivo. It can be degraded by a number of reactions, but under physiological conditions, NO concentrations are submicromolar, and it is the first-order reactions with superoxide and heme-containing proteins such as hemoglobin (Hb) and guanylate cyclase that should dominate its chemistry in vivo (3).

One of the important roles that NO plays in the body is the regulation of vascular smooth muscle tone. NO produced in the vascular endothelial cells can diffuse freely across cell membranes to the adjacent smooth muscle where it activates the enzyme soluble guanylate cyclase, leading to an increase in the intracellular cGMP concentration and to smooth muscle relaxation. The close proximity of the red blood cells (RBC) to the site of NO production and the fast consumption of NO by both oxy- and deoxyHb observed in vitro (10, 16) suggest, however, that a significant amount of NO will be scavenged by the blood. Thus it is unclear how much of the endothelium-derived NO is able to reach the smooth muscle where it needs to sustain physiologically significant concentrations for the activation of soluble guanylate cyclase.

A number of experimental and theoretical studies have been performed to investigate the diffusional spread of NO away from its site of production (5, 21, 24, 36, 40). Theoretical studies have recently been reviewed by Buerk (4). They represent a first generation of models that consider only transport of free NO. They do not account for preservation of NO-related bioactivity through the formation of more stable intermediates such as S-nitrosothiols. Theoretical simulations by Lancaster (21) showed that physiological amounts of Hb (2 mM) flowing in the lumen of a 20-µm arteriole scavenges significant amounts of NO, leading to a dramatic reduction of the NO concentration in the arteriolar smooth muscle. The result questioned earlier experimental observations that free NO is the endothelium-derived relaxing factor (12, 20, 26). Butler et al. (5) included in their theoretical model a layer free of RBCs next to the endothelium. They demonstrated that despite the significant scavenging of NO by Hb, a substantial amount of NO diffuses toward the smooth muscle; at least for vessels with diameters >160 µm that were examined. They also suggested that the reaction of NO with the Hb "packed" in RBCs might be slower than the reaction of NO with free Hb due to transport resistance through the membrane. Using a detailed mathematical model, Vaughn et al. (36) also suggested the importance of the RBC-free layer in the NO diffusion toward the smooth muscle; however, they concluded that the uptake of NO by RBCs has to be several orders of magnitude smaller than the uptake by an equivalent amount of free Hb in solution for the concentration in the smooth muscle to reach physiologically significant levels. It became obvious that a more detailed description of the NO uptake by RBCs must be incorporated in the modeling studies of NO diffusion.

Recently, two research groups performed experimental studies combined with theoretical analyses to acquire a more detailed description for the uptake of NO by RBCs. Liu et al. (22) measured the disappearance of NO in a suspension of RBCs in a phosphate-buffered solution with the use of an NO-sensitive electrode. Small concentrations of RBCs were utilized (three orders of magnitude less than blood). The disappearance of NO followed a first-order reaction rate with a half-life on the order of seconds. This corresponds to a reaction rate constant ~650 times less than that of free Hb. Utilizing a mathematical model for current flow around a microelectrode, Lui et al. (22) were able to reproduce their experimental observations. Thus they attributed the significant reduction in the reaction rate to external diffusion resistance in the transport of NO from the solution to the RBC membrane. Assuming that this resistance remains the same independent of hematocrit (Hct), they suggested a linear relationship between NO consumption rate and Hct and extrapolated their prediction to normal Hct.

Vaughn et al. (35) used the "competition experiment," where RBCs in a suspension with free Hb are competing for NO generated in a homogenous phase by an NO donor, to measure the NO uptake by RBCs at high Hct under conditions that should minimize the external diffusion resistance. Measurement of the extracellular methemoglobin (MetHb) concentration that is formed allows the estimation of the ratio of the rate of uptake of NO by the RBC and by free Hb and, thus the ratio of the reaction rate constants. Vaughn et al. (35) noted that the external diffusion resistance should decrease with increasing Hct due to a smaller plasma layer around each RBC. Their experimental data, however, suggested that the reaction rate for the RBCs plateaus for Hct >5%. They attributed this behavior to a significant resistance in transport through the RBC membrane or intracellular diffusion limitations, which become the rate-limiting step of NO uptake at higher Hct. In experiments performed at higher Hct (15.6%), changes in the NO donor or free Hb concentrations did not alter the rate of uptake by RBCs, providing additional indications that NO transport is not external diffusion limited, at least under these experimental conditions. The NO consumption by the RBCs was a 1,000 times less than that of an equivalent concentration of free Hb. In a subsequent study (34), they utilized a more detailed model for the analysis of the competition experiment. The model takes into consideration internal, external, and membrane diffusion limitations. The model was fitted to the experimental data for the estimation of membrane permeability or the intracellular reaction rate constant. The predicted value in either case was 2,000 times smaller than expected. Although the competition experiment could not distinguish between the two resistances, the fact that significantly reducing the intracellular Hb concentration did not affect the rate of uptake suggested that it is likely the membrane permeability that limits the NO transport. However, NO is a small and highly diffusible molecule that has been previously thought to have a high membrane permeability (3, 24, 30).

Recently, Huang et al. (17) proposed a mechanism to explain such a significant resistance to NO transport in the RBC membrane. They utilized the same competition experiment to estimate the rate of NO uptake by pretreated RBCs. The results suggested that altering the band 3 binding to cytoskeleton or altering metHb and denatured Hb binding to the RBC membrane or cytoskeleton, significantly alters NO uptake by the RBC. They concluded that RBC membrane- and cytoskeleton-associated NO-inert proteins provide a significant barrier for NO diffusion into the cell. In another experiment, changes in the viscosity of the solution did not alter the NO uptake rate significantly, suggesting negligible external diffusion limitation at least under the conditions of the competition experiment, thus providing additional indications that membrane resistance is the limiting factor for NO uptake.

Administration of extracellular Hb-based oxygen carriers (HBOCs) holds promise as an alternative to blood transfusion. The hypertensive effects often seen after administration, however, are considered a significant obstacle to the potential use of HBOCs (1, 29, 31-33, 38, 39). This phenomenon has been attributed to the scavenging of NO by the plasma-based Hb. Plasma-based Hb should be able to get closer to the endothelial cells in the lumen and can possibly extravasate to the space between the endothelial cells and the smooth muscle. In addition, the consumption of NO by a mixture containing RBCs and free Hb would be different from that of RBC alone.

The purpose of this paper is to provide an analytic description for the consumption of NO by the blood in the presence and absence of plasma-based Hb and for different levels of Hct. Such a description is needed for the development of detailed NO transport models in the presence of Hb-based blood substitutes. In the process, we examine previous hypotheses for the importance of the membrane and extracellular transport resistances for the uptake of NO by the RBCs. The theoretical predictions will be compared with previous analyses and available experimental data.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Model development. A spherically symmetric model was utilized to characterize the uptake of NO by an RBC as illustrated in Fig. 1. An RBC is assumed spherical and is surrounded by a plasma layer. NO diffuses through the plasma layer and the membrane of the RBC, reaching the intracellular region, where it reacts with the Hb through an irreversible, fast, first-order reaction. Hb is present in abundance inside the cell, and its concentration does not decrease significantly from the reaction with NO. Thus we assume that the RBC represents an infinite sink for NO. To characterize the NO uptake, we need to take into consideration both external and internal resistances to mass transfer as well as resistance for transport through the membrane.

View larger version (32K): [in this window] [in a new window]   Fig. 1.   Schematic representation of the model. A spherical red blood cell (RBC) is surrounded by a plasma layer with thickness that depends on hematocrit (Hct). Concentration at the outer boundary of the RBC is assumed constant. Simultaneous solution of the diffusion reaction equations in all three layers yields a concentration profile. Gradient of the concentration at the surface of the RBC is utilized to estimate the consumption rate of nitric oxide (NO) by the RBC. Integration of the profile yields the average NO concentration over the plasma layer or over the entire simulation volume. RR, RBC radius; RC, cytoplasm radius; RP, plasma radius; CP, concentration of outer boundary of plasma; RP, radius of plasma layer. See text and equations for other definitions of mathematical abbreviations.

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Plasma layer. The solution of Eq. 2 gives the concentration profile of NO in the plasma

(11)

where _P = () = 1 and _R =  (1). Differentiation of the solution at r = 1⁺ gives the rate of NO uptake by the RBC per unit RBC area (F_RBC)

(12a)

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(12c)

A more convenient description for F_RBC can be obtained by expressing the flux as a function of the average concentration of NO in the plasma layer (_pl)

(13a)

(13b)

(13c)

Replacing _P with the dimensionless average concentration estimated over the plasma layer (_pl) in Eq. 12 utilizing Eq. 13, we get

(14a)

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(14c)

RBC membrane. The solution of Eq. 3 utilizing Eqs. 6 and 7 provides the concentration profile in the membrane. Differentiation of the solution at r = 1 provides the flux at the outer boundary

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(15b)

where _C = () and P_m represents the membrane permeability, which is commonly used to describe the transport of species through a membrane.

Intracellular region. The solution of the differential mass balance in the intracellular region, Eq. 4 using Eqs. 7 and 10 yields

(16)

Differentiating Eq. 16 at r =  utilizing Eq. 9, we get an expression for F_RBC

(17a)

where

(17b)

Equations 14, 15, and 17 describe the flux into the RBC as a function of concentration gradients in the plasma, membrane, and cytoplasm. The three equations can be combined by adding the three in-series resistances as follows

(18a)

where

(18b)

The total uptake of NO per unit RBC volume will be

(19)

and the local consumption of NO per unit blood volume

(20)

The average concentration of NO in the RBC (membrane and intracellular) will be

(21a)

where

(21b)

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(21d)

Then the local (average) NO concentration C_NO will be

(22)

Replacing _pl in Eq. 20 gives

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where k_blood is the observed first-order rate constant of NO consumption in the blood. Note that because of the linearity of Eqs. 2-4, the calculated reaction rates k_RBC and k_blood are independent of the concentration C_P. The half-life of NO in the whole blood (t) and plasma (t) will be

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Note that the two definitions of NO half-life are equivalent for very dilute RBC solutions (Hct  0) or for negligible NO concentration in the membrane and cytosol of the RBC (g  0).

Parameter values. Values used in calculations are presented in Table 1. R_R can be estimated such as to conserve either the volume (90-98 µm³) or the surface area (130-144 µm²) of a human RBC (2, 11). Thus we examine a range of values for R_R between 2.8 and 3.38 µm. The D_pl was set to 3.3 × 10⁵ cm²/s at 37°C based on the data from Malinski et al. (24). D_pl at 25°C was assumed 2.6 × 10⁵ cm²/s based on the diffusivity of NO in water at 25°C. The D_cy should be decreased compared with the plasma due to the high concentration of Hb present. We set D_cy to half the value of D_pl (i.e., 1.6 × 10⁵ cm²/s at 37°C) based on the ratio of the extracellular and intracellular diffusivities for O₂ from experimental measurements (14, 28) and assuming a similar dependence for NO. Malinski et al. (24) suggested values for the diffusivity of NO in the lipophilic environment of a membrane (D_m) of 0.3 × 10⁵ cm²/s and a partition coefficient () of 6.5 for the membrane-water system, based on measurements performed on a 1-octanol-water system at 37°C. Denicola et al. (9) measured the diffusion coefficient of NO in the RBC plasma membrane (0.4 × 10⁵ cm²/s) and in liposomes (1.3 × 10⁵ cm²/s) at 20°C by utilizing a fluorescence quenching technique. Thus, based on value of 0.4 × 10⁵ cm²/s for D_m and a membrane thickness of ~7 nm, Eq. 15b suggests a P_m of ~40 cm/s. This value is in agreement with the value of 93 cm/s reported by Subczynski et al. (30). The value for P_m utilized by Vaughn et al. (34) to explain the competition experiment is 2,000 times smaller (0.041 cm/s).

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	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Model analysis. The solution of model equations for the reference values of parameters is presented in Fig. 2. The dimensionless concentration () is plotted as a function of dimensionless distance (r) from the center of the RBC. Control parameter values are utilized and simulations are performed for two different levels of Hct: 45% (Fig. 2A) and 15% (Fig. 2B). The average dimensionless concentration estimated over the plasma layer (_pl) or over the total plasma and RBC volume (_NO = C_NO/C_P) is also plotted. There is a discontinuity in the NO concentration profile at the RBC membrane due to the increased solubility of NO in the lipophilic environment of the membrane. The thickness of the plasma layer changes with Hct leading to changes in the NO uptake by the RBC.

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Dimensionless concentration as a function of dimensionless distance from the center of the RBC. Model simulations are performed for the reference parameter values and for two different Hct levels, 45% (A) and 15% (B). Dimensionless average plasma concentration (pl) and the dimensionless average concentration over the entire volume (NO = CNO/CP) are also presented. See text and equations for definitions of other mathematical abbreviations.

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View larger version (11K): [in this window] [in a new window]   Fig. 3.   Observed first-order reaction rate constant of NO consumption in the blood (kblood) as a function of RR (A), bimolecular reaction rate constant of NO with free hemoglobin (Hb) (koxy) (B), and membrane permeability (Pm) (C). Solid circles represent the reference parameter values. Other previously reported values for the parameters are also highlighted for reference (see text).

Comparison with experimental data. In Fig. 4 the model predictions are compared with previously reported measurements of NO consumption by RBCs. The erythrocytic NO consumption rate per RBC volume and per average plasma concentration (k_RBC) is plotted as a function of Hct. Experimental data from a dilute suspension of rat RBCs (Hct was more than 2,000 times less than normal) at 25°C are presented (solid circles) (22). The extrapolation to normal Hct proposed by Liu et al. (22) is also shown (dashed line). Note that the data and model by Liu et al. were presented in Ref. 22 on a per blood volume basis (Eq. 10 of Ref. 22) and have been converted in this figure on a per RBC volume by dividing with Hct. Thus their model predicts a constant k_RBC independent of Hct. Data from Carlsen and Comroe (6) are represented by a solid triangle. Our theoretical predictions are also shown as a solid line. To simulate the experimental conditions, we utilized a value for R_R of 2.44 µm to account for a smaller size of rat RBC (volume of 60 µm³), k of 106 µM¹ · s¹ (based on a k_oxy at 20°C of 89 µM¹ · s¹ and extrapolation to 25°C using a temperature coefficient of 1.4 per 10°C) and D_pl of 2.6 × 10⁵ cm²/s (Fig. 4). There is a close agreement between the models as the Hct approaches zero. At physiological Hct, however, our results differ from those of Liu et al. (22) by a factor of 6. Our prediction for the half-life of NO in blood at 30% Hct (5 × 10⁹ RBCs/ml) is 0.23 ms, which is significantly less than the estimate of Liu et al. The experimental data collected at very low Hct cannot be used to distinguish between the two models. Simulation using values for P_m and k_oxy of 0.04 cm/s and 25 µM¹ · s¹, respectively, is also presented. These values lead to significant underestimation of the experimental data of Liu et al. (22) and Carlsen and Comroe (6). For this value of P_m the effect of Hct on k_blood is minimal.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Model predictions for the rate of NO consumption per unit RBC volume and per average plasma concentration (kRBC) as a function of Hct. Model simulations (solid lines) are presented for the control values for Pm and k and for a "low" Pm and k (0.04 cm/s and 25 µM1 · s1, respectively). The experimental data (solid circles) and the model predictions (dashed line) of Liu et al. (22) are also presented. Simulations for both models are performed with diffusivity of coefficient of NO in plasma (Dpl) of 2.6 × 105 cm2/s and RR of 2.44 µm. Solid triangle present data from Carlsen and Comroe (6).

Analysis of the "competition experiment." In Fig. 5, we present results from the "competition experiment" (35). The analysis of the corresponding problem is presented in the APPENDIX. The ratio of k_RBC/(kC) is presented as a function of Hct (Fig. 5A) or C (Fig. 5B). Note that this is equivalent to the ratio of k_RBC/k_Hb in the studies of Vaughn et al. (34, 35). Simulations are performed utilizing the single cell model of Ref. 34 (see APPENDIX) and for different scenarios of parameter values. First, and in agreement with Ref. 34, we utilized a "low" P_m (0.04 cm/s) and a "low" k (25 µM¹ · s¹). We also perform simulations for a k of 106 µM¹ · s¹ and a P_m 1,000 times higher (40 cm/s). All simulations were performed for D_pl of 2.6 × 10⁵ cm²/s, R_R of 3.38 µm, first-order reaction rate constant (k_d) = ln (2)/6 h¹, C_NO donor of 10 µM, and C of 9 µM or Hct of 15.6%. The experimental results presented as solid circles are extracted from Figs. 3 and 4 of Ref. 35. The ratio k_RBC/(kC) is essentially constant and independent of either Hct or C when the "low" P_m is utilized. When the control value for P_m = 40 cm/s is utilized, a positive slope is observed in Fig. 5, A and B. The model can simulate satisfactorily the experimental data without the need for a 1,000 times reduction of P_m when k is set to 106 µM¹ · s¹ instead of 25 µM¹ · s¹. Figure 5C presents estimations for P_m utilizing the single cell model of Ref. 34 and Eq. A8 in the APPENDIX of this study. Parameter estimation is performed for a wide range of values for k (25-175 µM¹ · s¹) and for values of the ratio k_RBC/(k C) in the range 0.0006-0.0024. Simulations were performed for Hct = 15%, D_pl of 2.6 × 10⁵ cm²/s, R_R of 3.38 µm, C of 9 µM, k_d = ln(2)/6 h¹, and C_NO donor of 10 µM. For high k values a wide range of P_m values can produce ratios of k_RBC/(kC) that are in close agreement with the experimental measurements (34, 35). At low k values only a small range of low P_m values are in agreement with the experimental data. With typical values from competition experiments at 15.6% Hct, of k_RBC/(kC) in the order of 0.0012 ± 0.0001 (35), and expected k within the range of 30-110 µM¹ · s¹, Fig. 5C suggests acceptable values for P_m within the range of 0.1-40 cm/s. On the basis of a ratio of 0.0012, an empirical correlation was obtained that produces pairs of parameter values for P_m and k that satisfy the competition experiment over the above ranges of variation for the two parameters

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In Fig. 6 two independent experimental observations are simulated for different values of P_m. t at 0.0126% Hct is simulated utilizing Eq. 25, R_R of 2.44 µm, and D_pl of 2.6 × 10⁵ cm²/s. In addition, simulations of the competition experiment (Eq. A8) are also performed in an effort to simulate the change in k_RBC/(kC) after doubling the viscosity of the solution (Fig. 4 of Ref. 17). Simulations of the competition experiment are performed for Hct of 15%, D_pl of 2.6 × 10⁵ cm²/s, R_R of 3.38 µm, C of 9 µM, k_d = ln (2)/6 h¹, and C_NO donor of 10 µM. In the simulations at any given P_m, a k_oxy value that satisfies Eq. 26 is chosen and thus the pairs of P_m and k_oxy utilized are in agreement with a ratio of k_RBC(kC) of 0.0012. The results are compared with the experimental measurement (±SD) of 4.25 ± 0.2 s for t (22) and the 15 ± 6% observed change in k_RBC after increasing the viscosity twofold (17). The ranges of P_m values that can reproduce these experimental observations with accuracy no worse than twice the standard deviation of the measurement are highlighted. Because these ranges do not overlap, there are no values for P_m that would quantitatively explain both experiments.

View larger version (21K): [in this window] [in a new window]   Fig. 5.   Experimental results from the competition experiment (solid circles) reproduced from Figs. 3 and 4 of Vaughn et al. (35). The ratio of reaction rate constants of NO consumption by RBC and free Hb [kRBC/(kC)] is plotted as a function of Hct (A) and extracellular Hb (B) concentration. Simulations are performed utilizing the model of Vaughn et al. (34) and for three different scenarios of parameter values. First scenario includes values for Pm and k of 40 cm/s and 106 µM1 · s1, respectively. Values for the second scenario are 40 cm/s and 25 µM1 · s1, respectively, and values for the third scenario are 0.041 cm/s and 25 µM1 · s1. The following parameters values were utilized in all three scenarios: Dpl of 2.6 × 105 cm2/s, RR of 3.38 µm, kd of ln (2)/6 h1, CNO donor of 10 µM, and C of 9 µM or Hct of 15.6%. C: Eq. A8 in the APPENDIX is utilized to estimate Pm for different values for k = 25-175 µM1 · s1 and different ratios of kRBC/(kC) = 0.0006-0.0024. Same values were utilized as before for the rest of the parameters. The parameter values for Pm and k utilized in the three scenarios above are also highlighted for reference.

View larger version (16K): [in this window] [in a new window]   Fig. 6.   Two independent experimental observations are simulated for different values of Pm. t at 0.0126% Hct is simulated utilizing Eq. 25 and the following parameters: RR =2.44 µm and Dpl = 2.6 × 105 cm2/s. In addition, simulations of the "competition experiment" Eq. A8 are also performed in an effort to simulate the change in kRBC/(kC) after doubling the viscosity of the solution for Hct =15%, Dpl = 2.6 × 105 cm2/s, RR = 3.38 µm, C = 9 µM, kd = ln(2)/6 h1, and CNO donor = 10 µM. In the simulations at any given Pm a value for k that satisfies Eq. 26 is chosen. Experimental measurements (±SD) for t from Ref. 22 and for the percent change in kRBC after increasing the viscosity twofold from Ref. 17 are also shown.

NO consumption at physiological conditions. In Fig. 7A, predictions for k_blood at physiological temperature (k) is presented as a function of Hct. Different P_m values were utilized. For any given P_m value, the corresponding k_oxy at 25°C k was estimated utilizing Eq. 26. For the extrapolation of k_oxy at 37°C (k), a temperature factor of 1.4 per 10°C was used. The rest of the parameters were held at the reference values. At 45% Hct, predictions for k vary between 7.5 × 10² and 6.5 × 10³ s¹ when P_m changes between 0.1 and 40 cm/s. In Fig. 7B, k is compared with the rate of reaction of free Hb. The ratio HctCk/k is plotted as a function of P_m for different Hct. k values are shown in the secondary x-axis. For P_m values between 0.1 and 40 cm/s, k is 500-250 times less than the reaction with an equivalent concentration of free Hb.

View larger version (20K): [in this window] [in a new window]   Fig. 7.   Model predictions for the observed first-order reaction rate constant of NO in blood (37°C, 45% Hct), k (A), and the ratio of NO consumption by the blood and by an equivalent concentration of free Hb, as a function of Hct and Pm (B). For the simulations reference parameter values were utilized and k from Eq. 26 and extrapolation from 25° to 37°C using a temperature coefficient of 1.4 per 10°C. Predictions for the range of Pm values that satisfy the "competition experiment" (Fig. 5C) are highlighted.

NO consumption after HBOC administration. In Fig. 8, we investigate the effect of Hct and plasma-based Hb concentration on the NO consumption. The reference parameter values for R_R, k_oxy, and P_m are utilized. The rate constant (k_sol), for the NO consumption if the erythrocytic Hb was uniformly distributed in the solution (i.e., as if the cell were lysed) is also presented (estimated as the product of k_oxy with Hct). In the absence of plasma-based Hb, the consumption is a strong, nonlinear function of Hct. As Hct goes to zero, k_blood also approaches zero, similar to k_sol. In the presence of plasma-based Hb, the consumption is practically constant independent of Hct; k_blood becomes equal to the product of k_oxy and C. For plasma-based heme concentrations of 40 µM (7.7 × 10² g/dl of Hb), the total NO consumption is not much different from that in the absence of plasma-based Hb and 45% Hct. However, at a physiologically important plasma-based heme concentration of 3,000 µM (~5 g/dl of Hb), consumption is increased significantly. The ratio R_HBOC of the consumption rate in the presence of 5 g/dl of plasma-based Hb over the consumption in the absence of plasma-based Hb and normal Hct (i.e., 45% Hct) is presented on Fig. 8B as a function of k_oxy. Simulations are performed for P_m values that change with k_oxy according to Eq. 26 at 25°C and 37°C. For the latter case, in addition to Eq. 26, extrapolation of k_oxy values from 25°C to 37°C was utilized at any given P_m value, using a temperature coefficient of 1.4 per 10°C. The consumption of NO in the presence of 5 g/dl of plasma-based free-Hb is increased approximately two orders of magnitude relative to the physiological condition (45% Hct).

View larger version (18K): [in this window] [in a new window]   Fig. 8.   Effect of Hct and plasma-based Hb concentration on the NO consumption. A: observed kblood as a function of Hct in the presence and absence of plasma-based Hb (C = 0 µM, 40 µM, and 3,000 µM). The rate constant for the NO consumption if the erythrocytic Hb was uniformly distributed in the solution is also presented (ksol). B: ratio of NO consumption rate in the presence of 5 g/dl of free Hb over the consumption in the absence of free Hb at 45% Hct is presented as a function of koxy. Simulations are performed for different Pm values and at two different temperatures, 25°C and 37°C.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

A simplified spherically symmetric mathematical model was utilized to predict the NO consumption by RBCs. The results suggest that under the reference parameter values, which include a value for the RBC membrane permeability based on experimental measurements in lipid bilayers, extracellular diffusion is the rate-limiting step. The plasma layer surrounding each RBC and thus the extracellular diffusion resistance depends on Hct. This leads to a nonlinear dependence of the consumption rate on Hct. For human RBCs at 45% Hct, we predict a NO consumption rate constant on a per blood volume basis of 6.5 × 10³ s¹ or 0.7 µM¹ · s¹ on a per total heme concentration basis, and a plasma half-life of 0.1 ms. These predictions were based on the assumption of P_m equal to 40 cm/s, which is in agreement with the expected NO permeability in the lipid bilayer of the RBC membrane.

The results presented in Fig. 3 suggest that under the control conditions described above, the consumption of NO in the blood as predicted by the model is sensitive to the effective radius of the RBC but not to the intracellular reaction rate or the membrane permeability. The results suggest that the resistance to NO transport is dominated by the extracellular diffusion. Thus changes in the size and probably the shape of the RBC may alter the NO consumption rate. The values of k_oxy and P_m have to decrease more than an order of magnitude for the resistances in the two layers (i.e., intracellular and membrane) to have a significant effect in the NO uptake by the RBC. At the value proposed by Vaughn et al. (34) for P_m, however, significant resistance is attributed to the membrane. Most importantly, this value leads to a significant reduction (almost 20 times) of the consumption rate.

Huxley and Kutchai (18, 19) tested the hypothesis that extracellular diffusion is the rate-limiting step in the initial uptake of O₂ by RBCs in a stopped-flow apparatus. Their experimental findings suggested that 82-100% of the resistance in O₂ uptake could be attributed to extracellular diffusion limitations. Experimental studies for O₂ uptake by RBCs by Coin and Olson (8) are also consistent with a significant diffusion boundary layer. This boundary layer is a result of O₂ depletion in the area surrounding the RBC, faster than molecular diffusion or convective mixing can replenish it. NO has similar diffusivity with O₂, and Hb consumes it faster than O₂; thus a significant boundary layer should be present for NO as well. Huxley and Kutchai (18) utilized the Frossling correlation to predict a maximum value for the external diffusional resistance of oxygen uptake by an RBC. Assuming laminar flow, no extracellular reaction, and no interactions with neighboring RBCs (i.e., infinite thickness of the plasma layer), the correlation provides a description for the Sherwood number Sh = k_em 2R_R/D_pl in the form Sh = 2 + 0.6 Re^1/2Sc^1/3, where k_em is the external mass transfer coefficient, the Reynolds number (Re) is defined as a function of the particles slip velocity, and the Schmidt number (Sc) is defined as the ratio of kinematic viscosity and diffusivity. Estimates show that the transport is not significantly facilitated by convective mixing, yielding Sh = 2 (18). Thus we get an estimation for the k_em equal to D_pl/R_R. Applying this relationship for NO and assuming that the membrane and internal resistances are negligible compared with the external diffusion resistance, we can get an estimation for the consumption of NO by RBCs and an observed reaction rate in the blood

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where S_RBC and V_RBC represent the surface area and volume of RBC, respectively.

The model presented by Liu et al. (22) utilized these assumptions and equivalent equations, and they found close agreement with experimental results for the uptake of NO by rat RBCs at very low Hct (2,000 times smaller than normal). They concluded that the rate of uptake of NO by RBC is diffusion limited. The small difference (~25%) between their model and experimental data might be attributed to the values chosen for D_pl or R_R. They utilized a value of D_pl = 3,300 µm²/s based on the experimental data of Malinski et al. (24) conducted at 37°C. This value is in close agreement with the expected diffusivity of NO in water at 37°C. Here we utilized a D_pl of 2,600 µm²/s [the diffusivity of NO in water at 25°C, corresponding to the experiment (22)], the external resistance is increased and resulting in a closer fit of their experimental data (Fig. 4). Our analysis suggests that although these assumptions may be valid for such low Hct, extrapolation of these equations to higher Hct can lead to significant differences, i.e., sixfold at 30% Hct (Fig. 4). These differences originate from the simplifying assumption that the plasma layer around the RBC has an infinite thickness (which is not true for higher Hct), which leads to an overestimation of the external diffusion resistance. Our model utilizing the control parameter values suggests, in agreement with Liu et al. (22), that the external diffusion resistance limits the NO transport at low Hct. However, we also suggest that this resistance decreases with increasing Hct, and becomes negligible as Hct approaches one.

Vaughn et al. (34) suggested that P_m is much lower than the value expected based on the diffusivity and solubility of NO in lipid membranes. Their competition experiment predicts a 2,000 times lower P_m than the experimental value of Subczynski et al. (30). Under such a condition there will be significant resistance in the membrane even at very low Hct and the transport will not be dominated by external diffusion limitations. Such a reduction of P_m leads to a significant reduction of k_RBC at very low Hct, resulting in disagreement with the experimental data of Liu et al. (22) (Fig. 4).

Analysis of the "competition experiment." The disagreement illustrated in Fig. 4 between experimental measurements for the rate of NO uptake by RBCs and the value for P_m proposed by Vaughn et al. (34) motivated us to reexamine the analysis of the competition experiment. For the analysis we utilized the single cell model (34) to predict the ratio k_RBC/(kC) (see Eq. A8) in the APPENDIX. This is equivalent to the ratio of k_RBC/k_Hb in the studies of Vaughn et al. (34, 35). We simulated experimental data for the ratio of k_RBC/(kC) from Ref. 35 rather than the metHb production from Ref. 34, because these data were easily extracted from the figures, and according to Eq. A8, these data are independent of the NO production rate (and thus, independent of k_d, C_NO donor). The ratio k_RBC/(kC) was experimentally determined in Ref. 35 from the ratio of metHb formation in the presence and absence of RBC. In the process, the NO production rate is eliminated from the fitting equation, and the results are independent of the NO production rate. This is not the case when the metHb concentration is measured only in a system with RBCs. The formation of metHb and thus the parameter estimation results will depend on the NO release rate from the NO donor. Interestingly, utilizing a higher value for k_oxy, we were able to reproduce the experimental findings without the need for a 1,000 times decrease in P_m (Fig. 5, A and B). Most importantly, the value of 106 µM¹ · s¹ utilized in the simulations represents a reasonable estimation for k based on the recent results by Herold et al. (16) and extrapolation at 25°C. Figure 5C suggests that accurate determination of k is needed for an accurate estimation of P_m from the competition experiment. In addition, for high values of k, the ratio becomes insensitive to P_m, suggesting that the rate of NO uptake is not limited by transport through the RBC membrane. Thus, based on the uncertainty in the value of k, a wide range of P_m values (0.1-40 cm/s) can be in agreement with the experimental findings in the study by Vaughn et al. (35). Uncertainty regarding other model parameters such as D_pl and R_R also contributes, to a smaller degree, to the uncertainty in the estimation of P_m.

NO consumption at physiological conditions. Previous experimental data have suggested a reaction rate of NO with RBC 650-1,000 times smaller than the reaction with free Hb (22, 34, 35). These data, however, were collected under nonphysiological conditions (i.e., different temperature and presence of NO donor and free Hb) and/or at lower Hct, and thus they do not provide a direct measurement of k_blood. The theoretical model presented above was utilized for the interpretation of these data and extrapolation to the desired conditions.

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NO consumption after HBOC administration. The model predicts significant increase of the consumption rate of NO in the presence of plasma-based Hb relative to that of the blood. The consumption is essentially constant, independent of Hct. This can be attributed to significant NO scavenging in the plasma layer surrounding the RBC, when plasma-based Hb is present. At a physiological important plasma-based Hb concentration of 5 g/dl, we predict that the consumption of NO in the blood will be increased approximately two orders of magnitude or ~20-fold per gram per deciliter of Hb concentration in the plasma (Fig. 8B). The exact value of this increase will depend on parameters such as membrane permeability, temperature, and k_oxy. The nonmonotonic dependence of the ratio R_HBOC on k_oxy observed in Fig. 8B is attributed to the values for P_m and k_oxy utilized in the simulations, which were chosen to agree with experimental measurements (Eq. 26). The value of P_m in the simulations increased as k_oxy increased (Eq. 26). The increase in k_oxy leads to an increase in NO consumption in the presence of plasma based-Hb, whereas the increase in P_m leads to an increase in the rate of NO uptake by the blood in the absence of plasma-based Hb. The result is a ratio of NO consumption in the presence and absence of plasma-based Hb that change in a nonmonotonic fashion.

Model limitations. The model assumes spherical RBCs and neglects convective mixing facilitation of NO transport in the plasma layer surrounding the RBC. Both assumptions can affect the extracellular resistance and thus the NO uptake by the RBCs. We also assumed negligible consumption of NO inside the RBC membrane. Liu et al. (23) suggested 300 times more rapid reaction of NO with O₂ inside the hydrophobic environment of membranes. The small volume of the RBC membrane and the third-order reaction rate of NO with O₂, however, suggest that this assumption has only a small impact on the results of the transport calculations. In the development of the model's equations, we also assumed steady state, based on the high diffusivities and reaction rates and the small volume. We assumed that Hb consumes the NO through an irreversible reaction. There is increasing evidence, however, that Hb can preserve the NO bioactivity through the nitrosylation of a cystein residue in the -subunit of the Hb molecule and later release the NO molecule probably in the form of a S-nitrosothiol (15, 25, 27). At this point it is not known to what extent this mechanism can result in the release of free NO.

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