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Department of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore, Maryland 21205
Submitted 6 June 2003 ; accepted in final form 22 October 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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mathematical model; vascular regulation; microcirculation; diffusion; blood substitutes
In recent years, NO has attracted attention as a signal transduction molecule that mediates a variety of physiological responses in different systems. In the vasculature, NO was identified as EDRF and a key regulator of vascular tone and blood flow (29, 39, 62). In response to hemodynamic or agonist stimuli, vascular endothelial cells produce NO, which can diffuse across cell membranes to adjacent smooth muscle where it activates soluble GC (sGC), leading to an increase in the intracellular cGMP concentration and smooth muscle relaxation.
The close proximity of red blood cells (RBCs) to the site of NO production and the fast consumption of NO by both oxy- and deoxy-Hb (26, 36) suggest, however, that a significant amount of endothelium-derived NO will be scavenged by blood. Thus it is unclear how the endothelium-derived NO is able to reach the smooth muscle where it needs to sustain physiologically significant concentrations for the activation of sGC. A number of experimental and theoretical studies have been performed to investigate the diffusional spread of NO away from its site of production in the vascular endothelium in an attempt to answer this question (16, 49, 56, 79, 82). Several hypotheses have been proposed that account for a more efficient delivery of endothelium-derived NO to adjacent smooth muscle and for NO preservation or a reduced rate of deactivation by Hb in the blood (32, 38, 53, 63, 78, 79).
Theoretical studies of NO transport have recently been reviewed by Buerk (14). Theoretical simulations by Lancaster (49) have shown that physiological amounts of Hb (2 mM) flowing in the lumen of a 20-µm arteriole scavenge significant amounts of NO, leading to a dramatic reduction of the NO concentration in arteriolar smooth muscle. This result questioned the previous notion that free NO is EDRF (29, 39, 62). Butler et al. (16) and Vaughn et al. (79) included in their theoretical model a layer free of RBCs next to the endothelium that reduces the rate of NO scavenging by blood. They demonstrated that a substantial amount of NO can diffuse toward smooth muscle. Vaughn et al. (79) concluded, however, 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 smooth muscle to reach the physiologically significant levels of 250 nM. This critical value in their analysis was based on an estimate of NO concentration for the half-maximum activity of sGC provided by Stone and Marletta (70). Subsequent studies by this group and others have suggested, however, a smaller concentration of NO for the half-maximum activity of sGC in the range of 23–120 nM (7, 19, 66, 84). This should affect previous predictions for the effective diffusion distance of NO and the maximum allowable consumption rate of NO in the blood to maintain vascular tone (79).
The NO consumption rate in blood at physiological hematocrit (Hct) has not been measured. Carlsen and Comroe (17) were the first to measure the rate of NO uptake in dilute solutions of deoxyerythrocytes (Hct 
1%). More recently, Liu et al. (53) provided an estimate of the rate of NO uptake by RBCs. They measured NO concentration decay in a very dilute RBC solution using a microelectrode. Their estimate for NO consumption rate was 650 times less than that of free Hb. They assumed the same relative value for the consumption rate at physiological Hct, i.e., a linear dependence between the consumption rate and Hct or Hb concentration. They attributed this reduction to external diffusion limitations in the boundary layer surrounding the RBC (53, 55). Vaughn et al. (77, 78) also measured the rate of uptake of NO by RBCs in reduced Hct solutions (<15%) to be almost 1,000-fold slower than that by an equivalent concentration of free Hb using a "competition experiment." They attributed, however, this reduction to RBC membrane- and cytoskeleton-associated NO-inert proteins that provide a barrier for NO diffusion (38) and proposed an effective RBC membrane permeability for NO that was 1,000-fold smaller than previously thought.
In view of this conflicting evidence, the contribution of the RBC membrane and extracellular diffusion resistances to the rate of NO uptake by RBCs remains controversial (53, 55, 74). We have recently analyzed the available experimental data using a mathematical model and derived an expression for the rate of NO consumption in the blood as a function of parameters such as Hct, RBC membrane permeability, NO-Hb bimolecular rate constant, and RBC diameter (74). Our analysis suggests that 1) the predicted importance of the RBC membrane in a competition experiment depends strongly on the assumed bimolecular rate constant for the NO-Hb reaction; and 2) if the extracellular diffusion resistance is important, then the NO uptake rate by RBCs depends on Hct in a nonlinear fashion. We concluded that, based on all available experimental data, a somewhat faster NO consumption rate in the blood than previously reported is more likely (250–500 times slower than that of an equivalent concentration of free Hb, depending on the value of RBC membrane permeability).
The RBC-free layer adjacent to the endothelium and the reduced rate of NO by Hb "packed" in RBCs present possible mechanisms that allow free NO to escape scavenging by Hb and reach the smooth muscle in physiologically significant concentrations to induce vasorelaxation. An alternative hypothesis was presented by Stamler and colleagues (32, 41, 63), according to which Hb, instead of irreversibly consuming NO through conversion to nitrate, actually preserves it by preferentially binding to an unoccupied heme pocket and subsequent transfer of NO to a thiol group in a cysteine residue of the 
-chain (Cys93). NO bioactivity is then released during the allosteric transition of Hb from the oxygenated (R) to the deoxygenated (T) state by transfer of the NO group to acceptor thiols such as glutathione (41) or cysteine residues of RBC membrane-bound proteins (i.e., anion exchanger AE1) (63). This scenario provides a mechanism for regulated release of NO-related bioactivity by the RBCs at regions of high oxygen demand to induce vasodilation and increase blood flow; this hypothesis is critically discussed in Ref. 30.
These studies concentrated on the importance of intra-arteriole NO exchange and did not look in detail into NO transport/consumption in the perivascular region. We have presented theoretical results accounting for significant scavenging of NO by Mb in the tissue surrounding arterioles (45, 73) that can affect the NO concentration in smooth muscle. This result is consistent with experimental observations in isolated perfused hearts showing MetHb formation after an infusion of NO or bradykinin-induced release of NO (28). Mb knockout mice also showed increased vasodilatory sensitivity in response to NO release relative to wild-type mice. The above experimental observations prompted Brunori (13) to suggest a new role for Mb: in addition to serving as a reservoir for oxygen and facilitating its diffusion, Mb may play a role in preserving mitochondrial respiration by scavenging NO. A recent report by Pearce et al. (64), however, suggests cytochrome c oxidase and not Mb as the major route of NO deactivation in cardiac myocytes.
In addition to substrates such as Mb and cytochrome c oxidase, Hb in capillaries surrounding an arteriole might present a significant sink for arteriolar-derived NO. The rate of NO uptake by Hb in capillaries should depend on parameters such as capillary density, capillary Hct, and RBC membrane permeability. We formulated a mathematical model to provide a quantitative description for the rate of uptake of NO by RBCs flowing in a capillary as a function of the above parameters (75). This description is incorporated in the present study to provide a more detailed representation of extra-arteriolar NO consumption.
Most of the theoretical models that examine the spread of NO assume constant NO production and steady state. It is possible, however, that a transient production of NO occurs in response to agonist or hemodynamic stimulation. A burst-like NO production is often observed after activation of endothelial cells by Ca2+-elevating agonists; periodic bursts of intracellular free Ca2+ in response to a constant agonist concentration have been observed in a number of nonexcitable cell types including endothelial cells (8, 51, 58). The period of these oscillations varies from less than a second to a few minutes and in some cases is increased with ligand concentration. Such oscillations in the intracellular free Ca2+ concentration may result in bursts of NO production. Transient or sustained release of NO has been reported in response to shear stress. There is controversy regarding shear stress-induced Ca2+ transients in endothelial cells with some investigators reporting multiple Ca2+ transients (35, 37, 68, 69), whereas others report only irregular appearances or not at all (25, 59, 67). Ca2+ bursts appeared with duration of 10–15 s and a shear stress-dependent frequency on the order of 0.05–2 peaks/min (35, 37). It is possible, however, that shear stress regulates NO release in a Ca2+-independent fashion through phosporylation of eNOS resulting in a sustained basal NO production irrespective of the presence or absence of Ca2+ transients (4, 10, 27). Kutchan and Frangos (47) measured NO end-oxidation products (
and 
) released from endothelial cells exposed to laminar flow and reported transient Ca2+-dependent NO release on the onset of shear stress but a continuous basal release at constant shear stress levels. Buerk and Riva (15) observed spontaneous low-frequency NO oscillations in the cat optic nerve head, which they attributed to a natural variation in shear stress. On the other hand, Kanai et al. (42) measured NO release from endothelial cells exposed to constant shear stress with the use of NO-sensitive microelectrodes. They reported shear stress induced periodic Ca2+ transients and a concomitant release of NO. Thus for both agonist and hemodynamic stimuli an oscillatory pattern of NO formation might occur with a burst duration of a few seconds and a stimuli-dependent frequency on the order of tens of seconds to a few minutes.
sGC is the most well-established target of NO. The activation of the enzyme is thought to proceed through a two-step kinetic mechanism, according to which initial binding of NO to sGC results in partial activation of the enzyme through the formation of a six-coordinate nitrosyl intermediate. Subsequent conversion to a five-coordinate nitrosyl complex is thought to proceed through NO-dependent and -independent pathways (84). A NO concentration in the low nanomolar range is considered adequate for the activation of the enzyme, although a wide range of concentrations has been reported for half-maximum activity of the enzyme (23–250 nM) (5, 19, 66, 70, 84). The activation half-life is fast (usually a few seconds) and NO dependent (5, 19, 84). The in vitro sGC deactivation half-life is on the order of 10–100 min but can significantly decrease in the presence of NO scavengers such as thiols or Mb (12, 46, 57). In vivo, however, NO effects dissipate within 1–2 min after the termination of NO release, which suggests an in vivo sGC deactivation in <2 min (62).
The potential agreement between the duration and period of a transient NO release and the activation and deactivation half-life of sGC, respectively, suggests an alternative hypothesis for the mechanism that allows free NO to escape scavenging by Hb and induce relaxation of smooth muscle. In addition to diffusive barriers and preservation of NO-related bioactivity through the formation of nitrosothiols, transient bursting of NO with a duration large enough to activate sGC, followed by a period of basal or no release while sGC slowly deactivates, may provide a more efficient way for the maintenance of vascular tone.
In the present study, we tested the hypothesis that transient, burst-like NO production presents a way for NO to escape scavenging and facilitates efficient NO delivery to smooth muscle. We examined how transient conditions affect previous theoretical predictions for the ability of free NO to maintain vascular tone in the presence of NO scavengers such as Hb and Mb. To accomplish this objective, we developed a detailed mathematical model that describes NO transport in and around an arteriole. This model is an advancement over previously published models in that it incorporates a more detailed description of NO consumption in the lumen, smooth muscle, and extra-arteriolar region and transient evolution of NO. The model makes predictions for cGMP formation in smooth muscle under conditions of sustained and transient NO release.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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The unsteady reaction-diffusion equation is formulated to simulate NO transport in the radial direction in each of the regions. Axial and angular NO concentration gradients are assumed negligible due to symmetry in NO production, negligible Hb consumption along the arteriole, and large Damkohler and Peclet numbers (44)
 | (1) |
Continuity of concentration and conservation of mass at the interfaces yield the following boundary conditions
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| (4) |
Arteriolar lumen. In the core of the arteriole, the RBC-plasma solution is treated as a one phase medium. An effective first-order reaction rate constant is used to describe the consumption rate of NO in the region through the irreversible reaction with Hb
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The expression allows estimation of kblood as a function of local Hct, RBC effective radius (RR), RBC membrane permeability (Pm), NO-Hb bimolecular rate constant (kHb), NO diffusivity in RBCs and plasma (DRBC and Dpl, respectively), Hb concentration in RBCs (
), and concentration of free Hb in plasma (
).
NO is consumed in the cell-free layer through reaction with substrates in plasma, such as superoxide (
), thiols, or plasma-based Hb. An overall first-order reaction rate describes consumption in the cell-free region of the lumen
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Interstitial gap. The NO consumption rate in the interstitial gap between endothelial and smooth muscle cells is relative small. Significant consumption may occur in the presence of free Hb [i.e., after extravasation of a transfused Hb-based oxygen carrier (HBOC)]. An overall first-order reaction was used to approximate the consumption in the interstitial region
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Smooth muscle. In the smooth muscle, NO reacts with sGC. In the process sGC is activated, leading to cGMP formation from GTP. Zhao et al. (84) proposed a kinetic mechanism for the activation of sGC, which is shown in Fig. 2 [adapted from Condorelli and George (19)]. NO binds to the basal form of the enzyme (E1) and activates it through the formation of a six-coordinate partially active nitrosyl intermediate (E2). The reaction proceeds to the formation of the fully active five-coordinate enzyme (E3) through NO-dependent and -independent pathways. Condorelli and George (19) modified the above mechanism to include a first-order dissociation constant (kD) of NO from E3 to account for the effect of NO scavengers such as Hb, Mb, and thiols on the half-life of the active enzyme. According to this mechanism, we have the following expression for the consumption rates of NO, E1, E2, and E3
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 | (11) |
 | (12) |
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In a recent study, Ballou et al. (5) suggested that NO acts as a catalyst in the conversion of E2 and E3 and is not consumed in the process. This would suggest that the term in parentheses in Eq. 9 should be omitted. In any case, the impact of this term in the NO concentration profile is minimal.
Condorelli and George (19) analyzed this kinetic mechanism under well-mixed conditions. Here, we assume that all three forms of the enzyme can diffuse freely within the smooth muscle region and establish concentration gradients. The following reaction-diffusion equation is solved for all three forms of the enzyme simultaneously with Eq. 1
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We assume that the diffusion coefficient DGC is the same for all three forms of the enzyme. Zero flux boundary conditions are imposed at the two ends of the smooth muscle region as the enzyme is restricted inside the smooth muscle cells
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| (14) |
In this analysis, we assume for simplicity a single layer of smooth muscle cells, and thus the concentration Ei is continuous throughout the region. Initially, we assume that all sGC is present in the basal form [i.e., E1(t = 0, r) = Eo, where Eo is the total sGC concentration].
Integration provides the average concentration of each of the species in smooth muscle
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The cGMP formation rate at time t relative to the maximum possible formation rate can then be estimated as follows
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GCB = kGCB/kGC5 and GC6 = kGC6/kGC5. Note that the relative cGMP formation rate depends on the relative activity of the six- and five-coordinate nitrosyl forms of sGC with respect to the basal form (GC) and is independent of the turnover numbers, kGC.
Time-averaged relative cGMP production is considered as an index for the ability of NO to maintain vascular tone under different simulation scenarios
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Tissue. Reactions with heme-containing proteins such as Mb and cytochrome c oxidase should dominate the consumption of NO in tissue. A first-order reaction rate mechanism was used to describe the consumption of NO in nonperfused tissue
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Arteriole-derived NO may be consumed through the reaction with Hb flowing in the capillaries. On the other hand, capillary endothelium represents a potential source for NO. Thus a net production or consumption of NO is possible in the capillary-perfused tissue region. We (75) previously examined the exchange dynamic of NO around capillaries and derived a linear expression for the average NO exchange rate (RNO,pt); positive RNO,pt represents net consumption and a negative net production
 | (19) |
, , and tissue concentration of Mb (CMb). NO production by parenchymal tissue cells was assumed to be negligible in the present study, although it can be easily incorporated in the model. Numerical solution. We used a first-order central finite difference scheme for the spatial discretization of the partial differential equations. The resulting system of ordinary differential equations (ODEs) was solved using LSODES (65), which is numerical software intended for stiff and nonstiff systems of first-order ODEs with sparse Jacobian matrixes. Calculations were carried out until either a local relative accuracy of 10–5 or a local absolute accuracy of 10–4 nM was achieved in each point.
Parameter values. Values used in the calculations are presented in Table 1. Additional parameters used in the estimation of kblood and RNO,pt are reported in previous publications (74, 75). The diffusivity of NO in all regions was assumed to be the same and equal to 3.3 x 10–5 cm2·s–1 based on experimental measurements by Malinski et al. (56). A range of values has been previously reported for the reaction rate constant of NO with oxy-Hb (kHb) and oxy-Mb (kMb) (17, 18, 26, 36). For our simulations, we assumed kHb to be equal to 100 µM–1·s–1 and kMb to be equal to 55 µM–1·s–1. These values represent mean estimates from the available experimental data corrected for the temperature difference between in vitro and in vivo conditions.
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The overall reaction rate constants kpl and kig can be estimated from the product of kHb with the heme concentration in plasma () after HBOC transfusion and in the interstitial space (
) after extravasation of the transfused HBOC. kt is estimated from the product of kMb with CMb. In addition, we added a small value (1 s–1) to kpl, kig, and kt to account for the consumption of NO by other substrates present in the plasma or tissue. Such a value is justified based on the reaction rate of NO with [4,300 µM–1·s–1 (31)] and a concentration of in the subnanomolar range. Thus the consumption of NO in the plasma or tissue is dominated by the reaction with free Hb or Mb, and, in the absence of plasma-based Hb or Mb, the model accounts for consumption of NO in these regions through reactions with substrates such as . The value for the extravascular half-life of NO in the absence of Mb was within the range reported in Ref. 72. Here, however, we did not include a dependence of NO consumption on O2 concentration as in Ref. 72. Such a dependence of NO consumption on O2 concentration may result in cogradients for NO and O2 in the extravascular space and enhance tissue oxygen delivery (72).
The experimental study of Zhao et al. (84) suggested values for the kinetic parameters in Fig. 2 based on experimental data collected at 4°C. Condorelli and George (19) extrapolated these values to physiological temperature using the Arrhenius equation and assuming an activation energy of 55 kJ/mol based on literature data for similar reactions. This resulted in an order of magnitude higher value for kinetic constants at 37°C. The Marletta group (5) revisited their earlier experiments (84) and provided new estimates for these kinetic parameters. In the present study, we used values for k1, k2, and k3 from the study of Balou et al. (5) extrapolated to 37°C in a fashion similar to Ref. 19. The reverse kinetic constants k–1 and k–2 could not easily be estimated from Ref. 84. This is due to the high NO concentrations used in these experiments, which makes the reactions essentially irreversible. However, the experimental data suggest a maximum value for k–1. The almost complete conversion (i.e., >95%) of E1 to E2 under stoichiometric quantities of NO (590 nM) and E1 (470 nM) shown in Fig. 1 of Ref. 5 suggests that k–1 is less than k1(590–0.95·470)·0.05/0.95 = 7.5k1. We chose as a control parameter value for k–1 the upper limit of this range (15 s–1). In the absence of NO scavengers (kD = 0), k–2 is the rate-limiting step in E3 deactivation. We based our estimate of k–2 (0.001 s–1) on in vitro experimental data that suggest a half-life for the deactivation of nitrosyl sGC in the absence of NO scavengers of between 10 and 100 min at 37°C (12, 46, 57). The effect of NO on smooth muscle dissipates in 1–2 min, which suggests that in vivo NO-sGC deactivates in >1–2 min (62). Thus a kD value >0.008 s–1 is needed to provide a half-life of NO-sGC deactivation in agreement with in vivo observations (62). This parameter, in addition to the deactivation rate, significantly affects the NO concentration for half-maximal sGC activity. A range of values between 0.01 and 0.1 was examined.
The exact value of NO permeability of RBC membrane, Pm, remains at this point controversial. On the basis of the solubility and diffusivity of NO in lipid bilayers, Pm should be on the order of 40 cm·s–1 (22, 53, 56, 71), and this value was used as a control in the present study. However, recent studies suggested significant resistance to NO transport in the membrane of erythrocytes and a Pm value 1,000 times smaller (38, 77, 78). We (74) examined the effect of Pm on the predicted NO consumption rate by RBCs. We found that values in the range of 0.1–40 cm·s–1 are consistent with available experimental data if we incorporate the experimentally observed range of kHb. The effect of varying Pm within this range was examined.
The production rate of NO in the arteriolar wall is not well characterized. Information about the production in the capillary wall is also limited. Here, we used a control value of 2.65 x 10–3 nmol·cm–2·s–1 for both the luminal and abluminal side of the arteriolar wall endothelium based on experimental data by Malinski et al. (56, 79, 80) in the rabbit aorta and similar production rates per unit surface area for the capillary. In other words, in the absence of other data, we assumed that NO production per unit surface area by the arteriolar and capillary endothelium is the same as for large artery endothelium. In reality, the release rate of NO might be different due to different expression of the enzyme eNOS (3) and different physiological stimuli (shear stress, release of agonists). A wide range of values for the production rate was examined.
For the simulations below, we assumed control values of 45% for Hct in the lumen core of the arteriole and 33% in the capillaries. Simulations were performed for 1,500 capillaries/mm2 for the capillary density and in the presence and absence of 0.2 mM Mb in the tissue. A hypothetical scenario of HBOC transfusion was also examined with 15% and 10% Hct in the arteriole and capillaries, respectively, 3 mM free Hb in the plasma, and 100 µM free Hb in the interstitial gap.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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NO concentration profiles at steady state. Figure 4 shows several representative steady-state NO concentration profiles in and around a 50-µm-radius arteriole. For control parameter values, the average NO concentration in the smooth muscle is 83 nM. The presence of 0.2 mM Mb in the parenchymal tissue cells decreases the NO concentration to 28 nM. In the absence of extra-arteriolar NO scavengers such as superoxide, Mb, or Hb in surrounding capillaries, NO is consumed in the parenchymal space only through a reaction with molecular O2 (
). This reaction is slow, which results in an essentially flat concentration profile throughout the parenchymal tissue and yields an increase in the NO concentration in smooth muscle to 120 nM. A 10-fold acceleration of the reaction rate has been reported within the hydrophobic environment of lipid membranes (54). Such an increase has only a minor effect on NO concentration (data not shown). When we assumed a RBC membrane permeability of 0.1 cm/s instead of the control value of 40 cm/s, the NO concentration in smooth muscle increased to 122 nM. This was attributed to a change in kblood (from 756 to 6,400 s–1) as well as to a change in NO consumption rate by the capillaries (i.e., Spt = 1,145 nM/s and kpt = 13.6 s–1; control values of 482 nM/s and 25.9 s–1, respectively).
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In Fig. 4B, the effects of hemodilution and HBOC transfusion are examined. Reducing the Hct to 15% (Hctcap = 10%) resulted in an increase in CNO in the smooth muscle (140 nM). Transfusion of free Hb in plasma (3 mM) significantly reduced CNO from 140 to 8 nM. Extravasation of 100 µM free Hb to the interstitial gap further decreased the smooth muscle NO concentration to 2 nM. Hemodilution and HBOC transfusion affected both the NO gradient in the lumen of the arteriole as well as the equilibrium concentration in the parenchymal tissue away from the arteriole. The results are similar to those of our earlier study (45) except that here we used a more accurate model for capillary perfusion and NO consumption in blood and slightly different parameter values. The steady-state results question the significance of free Hb extravasation in generating vasoconstriction as the plasma-based Hb has already scavenged most of the NO.
Transient NO release. In Fig. 5, a representative analysis of transient simulation is presented. NO production follows a burst-like pattern modeled as a square wave with a duration (d) of 10 s and time period (T) of 30 s (Fig. 5A). The amplitude of the burst at the luminal and abluminal endothelium surface (Qt) is equal to the control steady-state release rate of NO (i.e., Qt = Q). Figure 5, B–D, presents model predictions for the NO concentration distribution (B), average enzyme concentrations in the smooth muscle (C), and relative cGMP formation rate (D) as a function of time. At the onset of the burst, the NO concentration increases and approaches a steady-state profile (like the control case in Fig. 4) within milliseconds (Fig. 5B). After the end of the burst, there is a rapid decrease of CNO toward a new steady state. Interestingly, the concentration gradient in the extra-arteriolar space is reversed as NO produced in the capillary endothelium diffuses toward the arteriole. In Fig. 5C, we trace the transition of sGC between the three different states (i.e., inactive E1, partially active E2, and fully active E3). Average concentrations in the smooth muscle are presented. After the increase in the NO concentration, E1 is rapidly converted to E2, followed by a slower conversion to E3. At the end of the burst (t = 10 s), 84% of the enzyme exists in its fully active form. After the arteriolar production shuts down, sGC deactivates slowly and thus sustains significant cGMP release even in the absence of arteriolar production. In response to a square wave of NO release, oscillations are established for enzyme concentrations after an initial transitional period. In this particular example, this is accomplished within the first 10 s. The cGMP relative formation rate during the established oscillations reaches a maximum value of 87% at the end of the burst and decreases down to 46% right before the next burst (t = 30 s). On the average, relative cGMP formation was 68% compared with 90% of the steady state; however, this was accomplished with only one-third of the steady-state NO release.
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Analytic solution. The relatively small consumption of NO by sGC and the fast establishment of a steady-state NO concentration profile at the onset and after the termination of a NO burst allowed us to decouple the NO diffusion from enzyme activation. An analytic approximation for the average relative cGMP formation rate can then be obtained as a function of enzyme kinetics, CNO in the smooth muscle, and T and d of the burst (see the APPENDIX). This expression was used to predict 
cGMP once CNO in the smooth muscle was known. This is particularly useful in scenarios of low CNO in smooth muscle where, unlike the example shown in Fig. 5, a significant time is required for the development of established oscillations. There is close agreement between the detailed numerical solution and analytic approximation (see the APPENDIX).
Behavioral analysis. In Fig. 6, the behavior of the model is examined through a series of model simulations using different parameter values (behavioral analysis). The model predicts cGMP for a wide range of values for the parameters 
, kD, d, and T. Dimensional stacking was used for presentation of the results in a color-coded composite two-dimensional plot. This represents an easy way to examine the models behavior as it varies with respect to more than two parameters (11). Apart from the parameters that define the transient of NO production (, d, and T), the choice of kD was suggested by the major effect of this parameter on the kinetic mechanism of sGC activation (Fig. 3C). We chose to indirectly vary Qt through changes in , d, and T
 | (20) |
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In this way, the average NO production rate remains constant along any horizontal line, which enabled us to examine the effectiveness of transient versus sustained NO release (the ratio of T/d =1 represents a sustained NO release).
The behavioral image shows results from 900 simulations. T/d and are the fast-varying dimensions and kD and d are the slow-varying dimensions in the horizontal and vertical axes, respectively. White lines mark the separation between slow-varying variables, and thus the squares that are formed contain simulations at constant kD and constant d. Control values were used for all other parameters, and simulations were performed in the presence of 0.2 mM Mb. The model predicts that significant activation of sGC is possible under both transient (T/d > 1) and sustained (T/d = 1) NO release. The exact level of activation, however, is strongly affected by the assumed values for kD and . Interestingly, the model predicts that a significantly smaller NO release rate (0.65 x 10–3 nmol·cm–2·s–1) than the one used in previous theoretical studies is capable of activating sGC (cGMP >50%). In the absence of Mb, the activation would be even higher. By examining the cGMP changes along a horizontal line within the same square, we can determine if a transient NO release (T/d > 1) is more efficient than a sustained one (T/d = 1). We found that this was often the case for kD = 0.01 or for low .
Sensitivity analysis. In Fig. 7A, cGMP is plotted in a color-coded two-dimensional plot. Simulations are presented from Fig. 6 for kD = 0.01 s–1 and d = 10 s. Figure 7, B and C, presents sensitivity images for the same scenarios of parameter values as in Fig. 7A. Absolute changes in the output of the model (cGMP) in response to a relative change in or T [i.e., relative sensitivity indexes 
and 
are plotted as a function of T/d and . In the top right corner of Fig. 7B, the relative sensitivity index of cGMP with respect to is <0.1. This suggests a minimal sensitivity of cGMP on and, as a result, insensitivity to changes in the amplitude of the burst Qt. Similarly, in the bottom left corner of Fig. 7C, the relative sensitivity index with respect to T is <0.1. This suggests that cGMP becomes insensitive to changes in T. (Note that a sensitivity index of 0.1 means that a 10% change in the parameter will change cGMP by only 1% of its maximal
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In Fig. 7, a line is defined by all points for which
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cGMP) with respect to T and are equal. The line divides Fig. 7A into two regions: one where cGMP formation is under frequency-dependent control (i.e., changes in T but not changes in Qt affect the formation rate), and one where it is under amplitude-dependent control (i.e., the opposite situation holds).
Efficacy of transient NO release. The potential for more efficient NO delivery under transient release is better demonstrated in Fig. 8. cGMP is estimated as a function of the ratio T/d, whereas the average NO release rate () is maintained the same by adjusting the amplitude of the burst (Qt) according to Eq. 20. Simulations were performed using Eq. A4 to predict cGMP once the NO concentration in the smooth muscle was known. The following scenario of parameter values for the kinetic mechanism of NO with sGC was examined: KM,VcGMP = 25 nM, t1/2,activ = 10 s, and t1/2deact = 90 s. A ratio of T/d = 1 denotes sustained NO release, whereas for T/d > 1 transient NO release is simulated as a square wave. Results are presented for different average NO release rates (). For low average release rates, transient NO release can significantly increase cGMP. For example, at = 0.1 pmol·cm–2·s–1, cGMP is 10% for sustained NO release and can increase to >35% for transient release with T/d > 6. For high NO release rates, there is a small increase in V a decrease in cGMP at low T/d, followed by cGMP at high T/d.
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HBOC transfusion and extravasation. To demonstrate the significance of frequency- vs. amplitude-dependent control of cGMP formation, we performed simulations for two different scenarios of parameter values represented as points in Fig. 7A. According to one scenario, the release rate was transient with T = 85 s, d = 10 s, and = 2.65 x 10–3 nmol·cm–2·s–1. In the second scenario, there was a sustained NO release rate with = 1.0 x 10–3 nmol·cm–2·s–1. Both of these points (scenarios) lay approximately on the same contour in Fig. 7A, which suggests that the cGMP formation rate was similar (80%). According to first scenario, however, cGMP formation is under frequency-dependent control, whereas the second scenario is under amplitude-dependent control. Figure 9 presents simulations under control Hct (45%) and in the presence and absence of 0.2 mM Mb. We also assumed negligible NO production by the capillary endothelium (Qcap = 0). The effect of hemodilution (15% Hct, 10% Hctcap), HBOC transfusion (
), and extravasation of transfused HBOC to the interstitial gap (
) were also examined. When the system operated under frequency-dependent control (open bars), the presence or absence of Mb in the tissue and hemodilution had little effect on cGMP. For the amplitude-dependent control scenario (solid bars), cGMP increased by 25% in the absence of Mb. Under the frequency-dependent control scenario, cGMP was maintained at 45% when 3 mM plasma-based Hb was present in the lumen of the arteriole. In contrast, under the amplitude-dependent control scenario, cGMP was reduced to 10% when free Hb was present. Extravasation further decreased cGMP in both cases. However, the decrease in the frequency-dependent control scenario, where a substantial formation rate was still maintained in the presence of plasma-based Hb, was much more pronounced than in the amplitude-dependent control scenario.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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Simulations of sGC activation based on the kinetic mechanism proposed by Zhao et al. (84) and using the best available values for the kinetic parameters reveal that a very small amount of NO (i.e., a few nanomolars) might be sufficient for sGC activation (Fig. 3). The value of the NO concentration for 50% maximum activity (KM,VcGMP) depends on a number of kinetic constants for which an accurate estimation is not yet available. For example, the experiments of Zhao et al. (84) were conducted under a relatively high NO concentration, which significantly favored the forward reactions (yielding essentially irreversible kinetics) and thus are not well suited for the estimation of the reverse reaction constants k–1 and k–2. The most significant uncertainty originates from kD. As Fig. 3B demonstrates, kD can significantly affect KM,VcGMP. Unfortunately, only a lower boundary for the value of kD is available based on the observation that NO effects dissipate in 1–2 min. Thus the predicted NO concentration necessary for sGC activation can be as low as 4 nM. Interestingly, there is an inverse effect of kD on KM,VcGMP and t1/2,deact: at high KM,VcGMP, t1/2,deact is low and vise versa. This may restrict the beneficial effects of a transient NO release relative to a sustained one as both a high KM,VcGMP and high t1/2,deact would be desirable.
Steady-state simulations revealed that the NO concentration in the smooth muscle is significantly affected not only by intra-arteriolar scavenging but also by extra-arteriolar consumption (Fig. 4). We identified consumption by Mb and exchange with capillaries but not reaction with sGC as the major determinants of NO concentration in smooth muscle. The presence of 0.2 mM Mb reduced the NO concentration in smooth muscle by >50%. Capillaries present 10 µm away from the arteriole resulted in a significant reduction of CNO in the smooth muscle. This was attributed to NO scavenging by the RBCs flowing in the capillaries. If we assumed a Pm value of 0.1 cm·s–1 [as recently suggested (77)], the NO concentration in smooth increased significantly. This was attributed to both a reduction in the intra-arteriolar and extra-arteriolar NO consumption. Interestingly, in this case, capillary-perfused tissue can maintain an average NO concentration of >80 nM and presents a potential NO source for the arteriolar smooth muscle. The sensitivity of the results to parameters such as Pm suggests that further experimentation is needed to better identify a number of important parameters before we are able to accurately predict the NO concentration profile around arterioles.
Can free NO escape scavenging by Hb and activate sGC? Simulations suggest that free NO may be able to escape scavenging by Hb and regulate vascular tone even without taking into account the recently proposed mechanisms (decreased RBC membrane permeability, preservation and transportation through formation of nitrosothiols). Our model suggests sufficient sGC activation even at a much lower sustained NO release rate than previously used (Fig. 6). The difference is attributed to the much lower KM,VcGMP (as low as few nanomolars) used here based on the latest available kinetic parameters for sGC activation (5). Significant uncertainty regarding the NO production and consumption rates as well as uncertainty in the level of NO required for sGC activation remains, and this does not allow us to unambiguously answer this question.
Transient versus sustained NO release. In response to a change in the NO release rate, the NO concentration profile approaches a new steady state relatively fast, within a few milliseconds (Fig. 5). This is attributed to a high diffusivity and a relative high consumption rate. In contrast, after an increase/decrease in the NO concentration in smooth muscle, sGC activation/deactivation proceeds much slower (Fig. 5C). The time required for enzyme activation is thought to be smaller than the time for deactivation. This generates favorable conditions for transient NO release. A transient NO production rate with a burst duration and period that corresponds to sGC activation and deactivation half-life, respectively, may improve the efficacy of NO in maintaining vascular tone. The simulations presented in Fig. 6 show that this is often the case under different scenarios of parameter values. The importance of a transient NO release becomes more pronounced as the balance among production rate Q, consumption rate R, and concentration for half-maximum activity KM,VcGMP becomes more unfavorable (i.e., Q decreases, R increases, and KM,VcGMP increases). Thus, as shown in Fig. 8, for KM,VcGMP = 25 nM and t1/2,deact = 90 s, transient NO release can increase cGMP more than threefold relative to sustained release at low levels of stimulation.
In Fig. 5B, the NO concentration gradient is reversed during the period of no arteriolar NO production, suggesting a flux of NO from parenchymal tissue to smooth muscle. This presents an interesting role for NO of parenchymal origin such as from capillaries or even from NO-producing parenchymal cells (43). Parenchymally derived NO may contribute to the maintenance of cGMP formation and delay sGC deactivation between NO bursts from the arteriolar wall.
Frequency-dependent control of cGMP formation. For a wide range of parameter values, simulations show that the activation of sGC by NO is under frequency-dependent control (Fig. 7, A–C). Whether frequency-rather than amplitude-dependent control of cGMP formation occurs depends on the balance between Q, R, and KM,VcGMP as well as how T and d compare with t1/2,deact and t1/2,activ, respectively. A favorable balance among Q, R, and KM,VcGMP (i.e., with Q increasing, R decreasing, and KM,VcGMP decreasing) and increased T/d promotes frequency-dependent control. This would suggest that it is the frequency of NO production and perhaps the frequency of Ca2+ oscillations in endothelial cells that determines vascular tone.
Many electrically nonexcitable cells including endothelial cells exhibit free calcium oscillations in response to a variety of stimuli. The physiological significance of these oscillations is currently under investigation. Woods et al. (83) and Berridge and Galione (9) have proposed that some physiological processes may be controlled by the frequency rather than the amplitude of Ca2+ sparks. This suggests that cells might be able to encode information in the frequency of Ca2+ oscillations. Recent experimental data provide the first experimental support for this hypothesis. The frequency of Ca2+ oscillations was shown to contribute to the sensitivity and efficiency of gene transcription (24) and calmodulin-dependent protein kinase II was able to decode the frequency of Ca2+ oscillations into distinct amount of enzymatic activity in an in vitro system (21). Here, along the same lines, we suggest that the interaction between Ca2+ and eNOS may result in establishing a role for NO as a transmitter of frequency-encoded information in Ca2+ oscillations from the endothelium to smooth muscle and for sGC as a decoder of the information in smooth muscle.
In a recent study, Condorelli and George (20) presented the first theoretical model of transient activation of sGC in smooth muscle by endothelium-derived NO in the bronchial circulation. They concluded that free NO should be able to activate sGC despite the rapid consumption by blood and that a transient NO release enhanced the efficacy of NO-modulated vascular tone. This previous analysis did not include a mechanism for transient activation of sGC; rather, the NO concentration was converted to the equivalent relative sGC activity level at steady state. For the system examined in the present study (systemic circulation), the NO concentration profile was quickly established (in milliseconds), and it is the kinetic mechanisms of NO binding to sGC that limit the activation-deactivation half-life and determine cGMP formation. In fact, even if the NO steady-state profile were to be established instantaneously, our results would not change significantly. In Ref. 19, pulsatile changes in blood flow and shear stress and cyclic behavior of muscular consumption were proposed as the causes for the periodicity in NO production. Here, we examined transient, burst-like NO release at lower frequencies, similar to those observed for Ca2+ oscillations in endothelial cells in response to agonist or hemodynamic stimulation.
Administration of HBOCs holds promise as an alternative to blood transfusion. The hypertensive effects often seen after administration, however, have been considered a significant obstacle to the use of HBOCs (1, 76, 81). This phenomenon has been attributed to the scavenging of NO by free Hb. NO consumption by free Hb in the lumen of an arteriole should be increased relative to the consumption by an equivalent concentration of Hb packed in RBCs (52, 53, 74, 78). In addition, free Hb may extravasate to the interstitial space, which would further decrease the NO concentration in arteriolar smooth muscle. The relative importance of these mechanisms is not fully understood. Most importantly, new generations of HBOCs are being developed aimed at reducing the vasoconstriction effect by reducing either the intraluminal NO consumption (modifying NO-Hb reactivity, encapsulation) or the degree of extravasation (polymerization, conjugation with larger molecules, encapsulation).
We (45) have previously presented a theoretical model that examines the effect of HBOC transfusion and extravasation on NO transport around arterioles. This model assumed sustained NO release and steady state. Simulations in this previous study and the present study (Figs. 4B and 9) predict that the presence of plasma-based Hb in physiologically important concentrations diminishes NO delivery to the smooth muscle. The simulations shown in Figs. 4B and 9 suggest that further extravasation of transfused HBOC is not as significant. The general notion, however, is that limiting HBOC extravasation by increasing the size of the transfused Hb molecule will diminish the vasopressor effect.
However, if NO release is transient and cGMP formation is under frequency-dependent control, the system is less sensitive to the presence of NO scavengers (Fig. 9). Thus the presence of Mb does not affect VcGMP significantly, and significant cGMP formation might be maintained after the administration of HBOCs (heme concentration in plasma of 3 mM). This scenario suggests a more important role for extravasation of transfused HBOC in generating vasoconstriction.
A frequencyrather than amplitude-dependent control of cGMP formation by NO may significantly affect the behavior of the system and thus the ability of NO to vasodilate. A transient model can provide a new theoretical framework to assist in interpretation of experimental data, analysis, and design and perhaps ultimately in drug development such as NO delivery drugs and blood substitutes. The theoretical simulations shown in Fig. 9 highlight the importance of delineating between the two modes of control.
Spontaneous rhythmic changes in vessel diameter (vasomotion) have been observed in a number of vascular beds. Although the complex waveforms of vasomotion are highly irregular and exhibit chaotic behavior, they often appear with a low-frequency periodicity (33). The observed frequencies of diameter change in vasomotion are in agreement with the frequencies of oscillatory NO production used in the present study. Thus one can speculate about a link between NO production and vasomotion. Such a link has not been established yet or at least not in all of vascular beds. There are reports in the literature with NO production promoting vasomotion in some tissues (40, 61), whereas vasomotion is independent or even suppressed by NO in others (14, 23, 34, 48, 60).
In conclusion, there are three major findings in the present study. First, free NO may be able to escape scavenging by Hb and regulate vascular tone. Interestingly, this may be accomplished even without taking into account decreased RBC membrane permeability or preservation and transportation of NO through interaction with thiols. Second, under certain conditions, a transient release of NO presents a more efficient way for the activation of sGC. Finally, simulations suggest that a frequency-rather than an amplitude-dependent control of cGMP formation is possible. This would suggest that it is the frequency of NO bursts and potentially the frequency of Ca2+ oscillations in endothelial cells that determines cGMP formation and regulates vascular tone and blood flow. The proposed hypothesis provides a new functional role for Ca2+ oscillations in endothelial cells. Further experimentation is needed to acquire a better description for the kinetics of sGC activation and NO production and consumption to delineate the two modes of control.
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APPENDIX |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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cGMP
In Fig. 5, VcGMP oscillations can be approximated by the following monoexponential relationships: activation (0 < t < d)
 | (A1) |
 | (A2) |
 |
 |
Equation 6 from Condorelli and George (19) can be used to estimate 
if the NO concentration CNO in the smooth muscle during the activation and deactivation phase is known
 | (A3) |
For k1 » k3, k3CNO » k2, and kD » k–2
 |
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Integration over a period gives
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| (A4) |
For the example examined in Fig. 5, the average NO concentration in the smooth muscle is 83 nM and falls to 6.2 nM after the burst. The predicted cGMP formation rate from equation is 70% compared with 68% from the numerical simulation.
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ACKNOWLEDGMENTS |
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This study was supported by National Heart, Lung, and Blood Institute Grant HL-18292.
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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and HO2: a pulse radiolysis study. Free Radic Biol Med 19: 505–510, 1995.[CrossRef][Web of Science][Medline]
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). B: simulation of Hb-based oxygen carrier transfusion and extravasation.
) and after extravasation of transfuced HBOCs to the interstitial gap (