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Mathematical modeling of the nitric oxide/cGMP pathway in the vascular smooth muscle cell

Departments of ¹Bioengineering and ²Electrical and Computer Engineering, Rice University; and Departments of ³Anesthesiology and ⁴Neurosurgery, Baylor College of Medicine, Houston, Texas

Submitted 8 March 2004 ; accepted in final form 5 April 2005

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
The nitric oxide (NO)/cGMP pathway in the vascular smooth muscle cell (VSMC) is an important cellular signaling system for the regulation of VSMC relaxation. We present a mathematical model to investigate the underlying mechanisms of this pathway. The model describes the flow of NO-driven signal transduction: NO activation of soluble guanylate cyclase (sGC), sGC- and phosphodiesterase-catalyzed cGMP production and degradation, cGMP-mediated regulation of protein targets including the Ca²⁺-activated K⁺ (K_Ca) channel, and the myosin contractile system. Model simulations reproduce major NO/cGMP-induced VSMC relaxation effects, including intracellular Ca²⁺ concentration reduction and Ca²⁺ desensitization of myosin phosphorylation and force generation. Using the model, we examine several testable principles. 1) Rapid sGC desensitization is caused by end-product cGMP feedback inhibition; a large fraction of the steady-state sGC population is in an inactivated intermediate state, and cGMP production is limited well below maximum. 2) NO activates the K_Ca channel with both cGMP-dependent and -independent mechanisms; moderate NO concentration affects the K_Ca via the cGMP-dependent pathway, whereas higher NO concentration is accommodated by a cGMP-independent mechanism. 3) Chronic NO synthase inhibition may cause underexpressions of K⁺ channels including inward rectifier and K_Ca channels. 4) Ca²⁺ desensitization of the contractile system is distinguished from Ca²⁺ sensitivity of myosin phosphorylation. The model integrates these interactions among the heterogeneous components of the NO signaling system and can serve as a general modeling framework for studying NO-mediated VSMC relaxation under various physiological and pathological conditions. New data can be readily incorporated into this framework for interpretation and possible modification and improvement of the model.

cell signaling; smooth muscle relaxation; calcium desensitization; signal transduction; integrative model

Endogenous NO is first synthesized in endothelial cells from L-arginine in the presence of the enzyme endothelial NO synthase (eNOS). It rapidly diffuses into neighboring VSMCs, where NO activates sGC, a heterodimeric hemoprotein, which catalyzes the conversion of GTP into cGMP. Acting as a second messenger, cGMP activates a family of serine/threonine protein kinase called cGMP-dependent protein kinases (PKGs), which has two primary effects on VSMC: 1) a reduction in cytosolic Ca²⁺ concentration ([Ca²⁺]_i) and 2) Ca²⁺ desensitization of the actin-myosin contractile system. Both effects ultimately lead to VSMC relaxation. Specific mechanisms of NO/cGMP-induced VSM relaxation have been reviewed elsewhere (8, 20, 21). The overall scenario of the endothelium-derived NO/cGMP pathway in the VSMC is shown schematically in Fig. 1.

View larger version (36K): [in this window] [in a new window]   Fig. 1. Schematic diagram of endothelium-derived smooth muscle relaxation via nitric oxide (NO)/cGMP pathway. In response to environmental neuronal, humoral, or mechanical stimuli (e.g., ACh, bradykinin, or shear stress), NO is synthesized in endothelial cells (EC) from L-arginine (L-Arg) by activated form of endothelial NO synthase. NO diffuses to neighboring vascular smooth muscle cells (VSMC), where NO activates soluble guanylate cyclase (sGC), which subsequently increases the intracellular cGMP production from GTP. B, basal form; 6c, 6-coordinate form; 5c, 5-coordinate form (fully activated). NO may directly (or by other pathway than producing cGMP) regulate certain target proteins [e.g., Ca2+-activated K+ (KCa) channel]. cGMP activates cGMP-dependent protein kinase (PKG), which regulates numerous target proteins, e.g., KCa channel current (IKCa), L-type Ca2+ channel current (ICaL), sarcolemma Ca2+-ATPase pump (ICaP), and myosin light chain phosphatase (MLCP), and leads to VSMC relaxation. cGMP is degraded into GMP by cyclic nucleotide phosphodiesterases (PDEs). Contractile kinetics shows Ca2+ and cGMP comediated MLC phosphorylation and cross-bridge attachment. M, fraction of the free form of myosin light chain; Mp, fraction of phosphorylated myosin, AMp, fraction of myosin attached to actin filament; AM, fraction of attached myosin cross-bridges but dephosphorylated, namely, latch state (16). Total myosin is conserved, i.e., M + Mp + AMp + AM = 1. Note that VSMC is connected with EC via myoendothelial gap junctions. R, hormone receptors on EC membrane; CaM, calmodulin; MLCK, myosin light chain kinase; Iup, sarcoplasmic reticulum Ca2+ uptake current; Irel, sarcoplasmic reticulum Ca2+ release current; IP3, inositol 1,4,5-trisphosphate.

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
sGC activation by NO. Figure 1 shows NO-activated transition of sGC from the basal state to the activated state (five coordinate, 5c). sGC is a heterodimer having - and -subunits with a heme in the center. NO stimulates the activity of sGC (300- to 700-fold) by binding to the heme of the enzyme. Recent studies suggest that this binding mechanism involves a two-step process (4, 9, 46). During the first step, NO binds to basal sGC to form an intermediate six-coordinate (6c) ferrous nitrosyl heme complex, which subsequently decays to a fully activated (5c) complex. The first step in NO binding to sGC is expressed as a reversible biochemical reaction process:

where E_b and E_6c represent sGC in the basal and intermediate (6c) forms, respectively, and k₁ and k_–1 are rate constants. In the second step, two parallel pathways, i.e., NO-dependent and NO-independent pathways, are involved in the E_6c-to-E_5c transition. The NO-independent process is considered a natural decay from state E_6c to E_5c, with the His105 residue being dissociated from the heme. This process can be described by first-order, irreversible kinetics:

with rate constant k₂.

For the NO-dependent pathway, a second NO binding site at the heme of sGC was proposed by Zhao et al. (46), which facilitates cleavage of the Fe-His105 bond. This process can be described as an enzymatic reaction based on traditional Michaelis-Menten kinetics (i.e., E + S ES P + E, where E is enzyme, S is substrate, and P is product). Because the reversible association and dissociation of NO with the isolated heme of sGC are rapid (the association rate constant was reported as 7.1 ± 2 x 10⁸ M^–1·s^–1 at 4°C in Ref. 46, and no intermediate was observed in the experiment), breakage of the His105 residue becomes the rate-limiting step in the process (the rate constant of this NO-dependent transition step was measured as 2.4 x 10⁵ M^–1·s^–1 at 4°C in Ref. 46), where the rate constant is three orders of magnitude smaller than the association constant for NO binding of the heme. Consequently, we assume that the NO-dependent pathway can be represented as a single-step irreversible process with rate constant k₃:

Another aspect of NO-sGC interaction is the dissociation of NO from the heme domain of sGC, i.e., the deactivation of E_5c and the recovery of the basal sGC condition. We describe this process according to a first-order, irreversible kinetics as:

where the turnover rate k₄ can potentially be modulated by several factors in vivo, including the sGC proximity to NO scavengers, concentrations of substrates, and [Ca²⁺] (22). Bellamy et al. (5) reported that sGC desensitization was observed in intact cells but not in lysed cells where substrates and products were diluted. In response to NO stimulation, the rate of cGMP production has a biphasic shape, with an initial rapid increase to the peak (3–4 s) followed by its later recovery to a lower level that matches the increasing degradation rate (within 2 min) as cGMP concentration reaches its steady state. Bellamy et al. (5) suggested that unknown sGC regulatory factors may exist that decrease sGC sensitivity to NO in the later phase of cGMP response and that these regulatory factors are active in response to the exogenous and endogenous presence of NO. Here, we propose that a local feedback loop, which senses the level of end-product cGMP, is the regulatory mechanism causing sGC desensitization. In our hypothesis, cGMP may not be the direct inhibitor for sGC activity but could be the second messenger initiating the negative feedback. We assume that NO dissociation from sGC and the recovery of E_5c to E_b is a cGMP-mediated process. In our model, the rate constant k₄ for the decay of E_5c back to E_b is expressed as:

(1)

where K₄ is a constant. Here, we assume that k₄ is proportional to the mth order of cGMP concentration, where m reflects the cGMP feedback strength.

On the basis of the above analysis of sGC kinetics, we describe the entire process with differential equations:

where E_b =[E_b]/[E₀], E_6c = [E_6c]/[E₀], and E_5c = [E_5c]/[E₀]. We assume that the total sGC concentration [E₀] is conserved, i.e., E_b + E_6c + E_5c = 1.

NO itself is consumed in vivo by NO scavengers (e.g., myoglobin, hemoglobin, and binding proteins other than sGC). We assume that NO is consumed immediately after dissociating from E_5c or NO leaves sGC in a form associated with other substances so that free NO is not liberated but instead taken to form other biochemical compounds. Consequently, the NO-sGC interaction results in a net loss of NO. Together with NO scavenging, the total NO consumption rate including NO binding to sGC may be approximated as a first-order kinetics. The dynamics of this process is expressed as:

where J_no is the endogenous or exogenous NO influx representing the physiological NO availability or NO analog introduced experimentally and t is time. Here, k_dno denotes the lumped NO consumption rate constant that reflects the activity of NO scavengers. Vaughn et al. (41) have estimated the rate of NO consumption in VSMC as 0.01 s^–1 for first-order kinetics.

cGMP production and degradation. Because the synthesis of cGMP begins after the catalytically active sGC E_5c is formed (46), we assume that cGMP production from GTP depends primarily on the degree of E_5c activity. The lifetime of cGMP is also regulated by the multiple isoforms of cyclic nucleotide phosphodiesterases (PDEs), which hydrolyze the cyclic nucleotide into GMP. The production and degradation of cGMP can both be modeled with Michaelis-Menten kinetics:

Assuming that the presence of GTP is abundant, cGMP production kinetics mainly depends on the sGC concentration. Study has shown that multiple types of PDE with different substrate affinities, subcellular locations, and kinetic properties (especially isoforms of PDE5 in VSMC) catalyze cGMP hydrolysis in VSMC (8). Here, we simply assume that PDE has a constant intracellular concentration and homogeneous catalytic affinity to its substrate cGMP. The balance equation for cGMP is written as:

(2)

where V_max,sGC represents the maximal rate of cGMP production when E_5c = 1, which implies that all available enzyme molecules participate in the catalysis, and V_max,pde is maximum cGMP hydrolysis rate. V_max,sGC is proportional to the total sGC concentration [E₀], which is conserved in our model. Hence, as shown in Eq. 2, the cGMP production and degradation rates are v_p = V_max,sGCE_5c and v_d = [cGMP]V_max,pde/(K_m,pde + [cGMP]), respectively. The Michaelis-Menten constant K_m,pde was measured in experimental studies by Turko et al. (39, 40) (see Table 1). Recent studies have shown that the basal level of PDE activity for cGMP hydrolysis is relatively low. However, this activity is greatly enhanced (up to 9- to 11-fold) after cGMP binding to its regulatory NH₂-terminal domain (GAF domain) (23, 33, 34). The PDE activation process by cGMP can be described as cGMP + PDE_inact PDE_act. Consequently, the concentration of active PDE depends on the level of cGMP, because in a Michaelis-Menten kinetics V_max is proportional to the total concentration of active PDE, which, we assume, is approximately proportional to cGMP concentration. This is a reasonable approximation if the PDE level is high compared with cGMP concentration. Therefore, we describe the maximum cGMP hydrolysis rate V_max,pde as being proportional to [cGMP]:

The cGMP regulatory effect can be rated by using a normalized function of cGMP concentration, R([cGMP]), which satisfies the following properties: R 0 when availability of cGMP is minimal, i.e., [cGMP] 0 (no effect), and R 1 when [cGMP] approaches its saturation level (maximal effect). Therefore, we consider cGMP as a molecular switch that turns the activities of its targets on and off (either all-or-none or concentration dependent). The regulatory function can be approximated by using the Hill equation:

(3)

where K_m and the Hill coefficient (n_H) represent the operating point and steepness of the switch, respectively. The above formulation constitutes an empirical steady-state approximation of the actual physiological mechanism of cGMP regulatory effects, which can be more accurately modeled only after biochemical details at the molecular level are unveiled by further experimental studies. Considering that cGMP binding to PKG and phosphorylation are relatively fast biochemical processes, we assume that it is reasonable to describe the downstream regulatory process with the Hill equation in Eq. 3. Consequently, the modulation of any target quantity Q in response to the change in cGMP is generally expressed by the following algebraic equation:

where Q_b represents the basal condition and Q_c represents the maximal change that can be induced by cGMP. Q can represent a variety of physical quantities including an ionic channel conductance, an enzymatic reaction rate constant, or an electrophysiological term (e.g., half-activation voltage) in vivo. In addition, both Q_b and Q_c can fluctuate as physiological conditions change.

Ca²⁺ entry reduction. Intracellular Ca²⁺ is modulated by several NO-mediated ion channels on the VSMC membrane. Here, we only consider the major channel target for cGMP: the large-conductance K_Ca channel. Activation of the K_Ca channel by cGMP induces membrane hyperpolarization, which subsequently deactivates the voltage-gated Ca²⁺ channel, shutting the inward Ca²⁺ flux and bringing about relaxation. Studies by Onoue and Katusic (28) on canine middle cerebral arteries indicate that VSMC relaxation to NO via activation of K_Ca channels may involve both cGMP-dependent and cGMP-independent pathways. A patch-clamp study by Gerzanich et al. (14) shows similar results in rat cerebral arteries.

In this study, we assume that NO regulates K_Ca channels via both the cGMP-dependent and cGMP-independent pathways. Specific activation of the K_Ca channel may be due to phosphorylation of channel proteins or its regulatory proteins (8). Studies by Zhou et al. (47) suggest that PKG stimulates the activity of two isoforms of the K_Ca channel proteins, which results in a shift of the voltage dependence of the single-channel open probability (P_o) toward more negative potentials. Other studies (31, 36) report similar cGMP-induced decreases in the voltage and Ca²⁺ thresholds of I_KCa. Therefore, we have chosen to model the regulatory effect of NO/cGMP on voltage dependence of the channel P_o described as:

where V_m denotes the membrane potential, V is the half-activation voltage, and S_KCa is the steepness factor. In our model, V is the major regulatory target for Ca²⁺, NO, and cGMP. The above P_o-V_m relationship can be shifted by modulating V. We hypothesize that NO exerts its effect by shifting the P_o-V_m relationship via both cGMP-dependent and cGMP-independent mechanisms. This is expressed as:

(4)

where the linear relationship between V and the logarithm of [Ca²⁺]_i (V_Ca and V_b) is given in our original VSMC model (42). Single-channel studies by Robertson et al. (31) on VSMC in rabbit cerebral arteries and Stockand and Sansom (36) on human mesangial cells found that V was shifted by cGMP-activated PKG toward a more negative voltage range, whereas the steepness factor S_KCa remained unchanged.

Ca²⁺ desensitization of MLC phosphorylation. Smooth muscle contraction and relaxation directly depend on actin-myosin interaction and are regulated by two competing processes: phosphorylation of the regulatory 20-kDa MLC by MLCK and dephosphorylation by MLCP. The activity of MLCK is a function of the concentration of the Ca²⁺-calmodulin complex (CaCM), whereas MLCP can be activated directly by cGMP/PKG. MLC activation and subsequent contractile force generation can be regulated by balancing the activities of the two enzymes MLCK and MLCP, i.e., the balance between MLC phosphorylation and dephosphorylation (20). It is well established that by activating MLCP, PKG phosphorylates the M110 regulatory subunit of MLCP and/or an MLCP inhibitor, decreases the MLC phosphorylation level, and desensitizes the responses of contractile system to Ca²⁺, without affecting MLCK activity. This causes the inhibition of MLC phosphorylation without altering [Ca²⁺]_i (8, 19, 26). Increased MLCP activity shifts the balance between two MLC phosphorylation regulatory enzymes, MLCK and MLCP, which leads to a contractile system less responsive to Ca²⁺.

In our previous study of the VSMC (42), we modified a kinetic model of MLC phosphorylation and cross-bridge formation originally developed by Hai and Murphy (16). The four-state model of contractile system kinetics is illustrated in Fig. 1, in which MLCK is regulated by the CaCM. Therefore, MLC phosphorylation was only a Ca²⁺-dependent process. In this study, we associate the MLC dephosphorylation rate constant with cGMP concentration to model the cGMP effect on MLC activity:

where k is the basal MLC dephosphorylation rate constant and k is a first-order rate constant for cGMP-regulated MLC dephosphorylation. The rate constant of MLC phosphorylation is Ca²⁺ dependent and defined as k_mlck = _mlck[Ca²⁺]. If we approximate the four-state myosin phosphorylation model (Fig. 1) with a reduced two-state phosphorylation kinetics:

(with the conservation constraint M + M_p = 1), we have, at equilibrium,

Ca²⁺ desensitization of the VSM contractile system is widely interpreted as a reduction of MLC phosphorylation at the same or a comparable Ca²⁺ level. The Ca²⁺ sensitivity of MLC phosphorylation can be formally defined as a nondimensional quantity _Ca, the magnitude of the relative change of MLC phosphorylation with regard to the relative change of [Ca²⁺]_i:

(5)

where K_ds = k_mlck/k_mlcp is the desensitization factor for the VSMC contractile system, which reflects the competing roles of MLCK and MLCP in regulating the MLC phosphorylation that is comodulated by Ca²⁺ and cGMP. The term is the order of Ca²⁺ influence on MLC phosphorylation. According to the above definition, the value of _Ca indicates Ca²⁺ sensitivity of MLC phosphorylation at the corresponding Ca²⁺ level. In the four-state model, the phosphorylation level (P) and the relative force (f) generated by cross-bridge interaction are calculated as P = M_p + AM_p and f = AM_p + AM, respectively, where M is the free form of MLC, M_p is the fraction of phosphorylated myosin, AM_p is the fraction of myosin attatched to the actin filament, and AM is the fraction of attached myosin cross-bridges but dephosphorylated.

Model protocol and parameter assignments. Model simulations are undertaken at a temperature of 20°C with extracellular ionic concentrations held constant: extracellular Na⁺ concentration ([Na⁺]_o) = 140 mM, extracellular K⁺ concentration ([K⁺]_o) = 5.0 mM, and extracellular Ca²⁺ concentration ([Ca²⁺]_o) = 2.0 mM. Cell volume is constant as 1 pl. Model parameters are listed in Table 1. VSMC precontraction is induced by one of two methods: 1) increasing membrane Ca²⁺ permeability by increasing L-type Ca²⁺ channel conductance g_CaL, which simulates the effect of substances such as UTP, and 2) increasing the Ca²⁺ sensitivity of contractile apparatus by increasing the MLC phosphorylation rate _mlck, which simulates the effects of phenylephrine (PE). Equations for modeling the four-state MLC phosphorylation and contractile kinetics are given in Eq. 20 in Ref. 42. All other model equations, parameters, and initial conditions of VSMC membrane channels can be found in the APPENDIX of Ref. 42.

The entire NO/cGMP signaling pathway is described by a system of deterministic ordinary differential equations programmed with MATLAB and solved numerically on a personal computer with an Intel-based processor. The numerical values of the majority of model parameters assigned are based on parameters adopted from the published literature, and the rest are identified within physiological ranges with a nonlinear least-square optimization technique by fitting reported experimental data (see Table 1).

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
sGC desensitization: a local feedback inhibition. NO-induced VSMC relaxation becomes deactivated during the latter phase of a short-term exposure of VSMC to NO (in seconds) (5, 13). Such a nitrate tolerance effect may be due to reduced sGC sensitivity to NO. This suggests that cellular regulation tends to stabilize the NO/cGMP pathway within a relatively wide range of NO availability and that sGC-activated cGMP accumulation and PDE-mediated degradation are two mechanisms that may regulate a variety of patterns of cGMP responses to NO. Changes in sGC sensitivity to NO indicate an immediate local feedback inhibition while the total sGC is maintained constant. As described in METHODS, we assume that sGC desensitization is primarily caused by end-product cGMP feedback inhibition.

Figure 2 shows the sGC desensitization effect with linear feedback (m = 1, Eq. 1) under conditions of constant NO stimulation (220 nM, as indicated in Ref. 5). Figure 2A shows that the model simulation reproduces the measured cGMP accumulation data (see Fig. 3b in Ref. 5). The cGMP production rate exhibits a biphasic time response similar to the analysis by Bellamy et al. (5): v_p sharply increases to its peak in <5 s and subsequently drops to a significantly lower level in 1 min (Fig. 2B), which indicates rapidly reduced sGC activity. As shown in Fig. 2C, the major fraction of the sGC population (>50%) is shifted to the intermediate form E_6c after a quick transient in E_5c, peaking in <5 s. This accumulation of E_6c is due to 1) an enhanced E_5c decay to E_b mediated by cGMP and 2) fast NO binding to E_b (i.e., high affinity of sGC for NO), which competes with the rate-limiting process converting E_6c to E_5c. In Fig. 2C, a low E_5c suggests that the steady-state cGMP level is well below the maximum production capacity.

View larger version (19K): [in this window] [in a new window]   Fig. 2. sGC desensitization by cGMP. A: comparison between model-generated simulation with cGMP accumulation data from Bellamy et al. (Ref. 5, Fig. 3b). [cGMP], cGMP concentration. B: rates of cGMP production and degradation (vp and vd, respectively). C: transients of all species of sGC. Exogenous NO source is simulated as a holding 220 nM as indicated in Ref. 5.

View larger version (24K): [in this window] [in a new window]   Fig. 3. Different orders of cGMP feedback. A and B: without cGMP feedback (m = 0). A: cGMP synthesis and degradation rates vp and vd. B: responses of fractional populations of 3 sGC forms. C and D: second-order cGMP feedback (m = 2). Exogenous NO source is simulated at a holding 220 nM concentration as indicated in Ref. 5.

View larger version (14K): [in this window] [in a new window]   Fig. 4. NO-cGMP and NO-E5c relationships. A: comparison of NO-induced cGMP production with cGMP feedback inhibition (m = 1) and without feedback (m = 0). B: comparison of E5c levels.

The quantities V_Ca, V_cGMP, and V_b in Eq. 4 are determined based on data from the study of Stockand and Sansom (36), in which cGMP-dependent PKG shifted V from 42 mV at control level to –34 mV at [Ca²⁺]_i = 1.0 µM. Figure 5, A and B, shows the model-generated P_o vs. V_m and P_o vs. [Ca²⁺]_i relationships, which compare favorably to measured data. This experiment consists of an all-or-none testing of the cGMP effect, in which the cGMP effect is tested at a saturated cGMP level with R_cGMP(K_m,cGMP,n_H,cGMP) = 1 without involving the cGMP-independent NO mechanism (R_NO in Eq. 4 set to 0).

View larger version (16K): [in this window] [in a new window]   Fig. 5. Effects of cGMP on the relationships between open probability (Po), cytosolic Ca2+ concentration ([Ca2+]i), and membrane potential (Vm) in the KCa channel. Under the conditions [Ca2+]i = 1.0 µM (A) and Vm = 40 mV (B), the model-generated data agree closely with measurements under both control conditions () and after cGMP saturation (). Data from Stockhand and Sansom (36).

View larger version (13K): [in this window] [in a new window]   Fig. 6. NO-induced cGMP-independent VSMC relaxation under control and 10% and 1% sGC activity conditions. A and B: with NO-directed activation of KCa channel. A: MLC phosphorylation (Mp + AMp) B: [Ca2+]i. C and D: without NO-directed effect. Preconstrictions were induced by a 10-fold increase in the L-type Ca2+ channel conductance gCaL. 1H-[1,2,4]oxadiazolo-[4,3-a]quinoxalin-1-one (ODQ) inhibition effect on sGC activity is simulated by reducing the enzyme-catalyzing factor to 10% and 1% of its control level.

Membrane depolarization can be induced by elevating [K⁺]_o as a result of reduced outward K⁺ current because of the reduced K⁺ gradient across the membrane and the equilibrium potential for K⁺ according to the Nernst equation. However, elevation of [K⁺]_o affects currents through all types of K⁺ channels including the K_Ca channel, the inward rectifier (K_i) channel, and the background K⁺ channel. Model simulations shown in Fig. 7 demonstrate this phenomenon, as well as the role played by I_KCa and K_i channel current (I_Ki) (Fig. 7, B and C). The log-linear relationship between V_m and [K⁺]_o shown in Fig. 7A is qualitatively consistent with experimental measurement of the V_m-[K⁺]_o relationship by Bratz et al. (7) in wild-type and NOS inhibitor-treated rats. The underlying currents, however, exhibit complex and nonlinear patterns. A reduction of inward rectifier conductance by 50% causes an increase in basal membrane potential and a decrease in the slope of the V_m-[K⁺]_o relationship (Fig. 7A). Same-magnitude reduction of K_Ca channel expression induces a much smaller depolarization effect (simulation not shown) because of the low activity of K_Ca channels at basal NO and Ca²⁺ levels. This result suggests that a reduced expression of the K_i channel caused by long-term NOS inhibition may be a more effective way to cause membrane depolarization than activating the K_Ca channel. The multiphasic characteristics of I_KCa, I_CaL, and [Ca²⁺]_i shown in Fig. 7, B, D, and E, are caused by the voltage-dependent activation and inactivation windows of the L-type Ca²⁺ channel operated in the voltage range between –30 and –20 mV (see Fig. 3 in Ref. 42).

View larger version (15K): [in this window] [in a new window]   Fig. 7. Basal NO influences on K+ currents and membrane potentials under control conditions (solid lines) and inward rectifier current (IKi) reduced by 50% (dashed lines). A: membrane depolarization induced by increased extracellular K+ concentration ([K+]o). B: Ca2+-activated IKCa. C: IKi. D: [Ca2+]i. E: ICaL.

Figure 8, A and B, shows comparisons between model-produced results and measured data (19). The contractile system responds to cGMP by shifting the steady-state Ca²⁺-force and Ca²⁺-MLC phosphorylation relationships into a range of higher Ca²⁺ levels. The _Ca-[Ca²⁺]_i curve as shown in Fig. 8C shifts toward the range of higher Ca²⁺ levels after the application of cGMP. The curve shifting in Fig. 8C represents the actual Ca²⁺ desensitization effect. The contractile system is least sensitive (minimal _Ca) to relative change in Ca²⁺ at saturated Ca²⁺ levels and most sensitive (maximum _Ca) at minimal Ca²⁺ levels. It should be noted that MLC phosphorylation becomes more sensitive to [Ca²⁺]_i variations when cGMP concentration increases and the contractile system is, however, desensitized (shown as decreases in relative force and MLC phosphorylation in Fig. 8, A and B). Here, we can see the conceptual difference between the Ca²⁺ sensitivity of MLC phosphorylation (quantitatively represented by _Ca) and the Ca²⁺ desensitization/sensitization of the contractile system. The term "Ca²⁺ desensitization" used in most of the literature refers to MLC phosphorylation and force reduction at the same Ca²⁺ level, whereas the term "Ca²⁺ sensitivity" indicates changes of MLC phosphorylation induced by Ca²⁺ variations.

View larger version (14K): [in this window] [in a new window]   Fig. 8. Ca2+ desensitization effects induced by cGMP and Ca2+ sensitivity of MLC phosphorylation. A: relative force Rf. B: MLC phosphorylation. C: Ca2+ sensitivity (Ca) under control condition [R(Km,mlcp,nH,mlcp) = 0, where nH is Hill coefficient] and after application of cGMP analog 8-bromoguanosine 3',5'-cyclic monophosphate (8-BrcGMP) at 10 µM [R(Km,mlcp,nH,mlcp) = 1] in the model because 8-BrcGMP reached the maximum regulatory effect at 10 µM in the experiments by Lee et al. (19). Model simulations, solid lines; , data in control condition; , with cGMP. Dashed line, R(Km,mlcp,nH,mlcp) = 0.33; dashed-dotted line, R(Km,mlcp,nH,mlcp) = 0.67.

	DISCUSSION

TOP ABSTRACT METHODS RESULTS DISCUSSION GRANTS REFERENCES

The signal carried by NO is detected by sGC in terms of a ligand-receptor interaction process with two-step sGC activation kinetics (Fig. 1). The model structure adopts the scheme proposed by Zhao et al. (46). To interpret the measured data of Bellamy et al. (5), we proposed a cGMP feedback-controlled sGC decay from activated form E_5c to basal form E_b. The model predicts that the intermediate E_6c is the dominant steady-state species of sGC under physiological NO stimulation (Fig. 2). This model prediction suggests that the sGC desensitization by cGMP feedback may limit or stabilize cGMP production within a wide range of [NO] (Figs. 3B and 4), which may contribute to the robustness of the VSMC response to small NO perturbations. Recovery of sGC desensitization is a much slower process that was completed in 10 min, as reported by Bellamy et al. (5). In our model, the recovery rate of desensitized sGC to control level directly depends on the rate of the cGMP degradation (Eq. 1). We showed that the k_pde value varies under different orders of cGMP feedback inhibition of sGC to produce the same NO-induced cGMP accumulation responses (see Table 2), which determine different cGMP degradation rates. Therefore, cGMP feedback order m in our model may be resolved by comparing the time constant of cGMP degradation to that of sGC recovery from desensitization.

cGMP regulation is a complex mechanism. Increasing amounts of experimental evidence suggest that basal PDE activity for cGMP hydrolysis is at a relatively low level. It can be activated (up to 9- to 11-fold) after cGMP binding to its regulatory NH₂-terminal domain (so-called GAF domain) and further enhanced by PKG phosphorylation (23, 33, 34). On the other hand, it has been suggested that sGC can be deactivated by PKG-dependent phosphorylation, which provides another negative-feedback inhibition to tune down cGMP production (27). Studies by Mullershausen et al. (24, 25) showed that cGMP can have a transient response to sustained NO and drop further to a lower level in a later phase. Such a biphasic cGMP response indicates that the rate of cGMP synthesis is outcompeted by the degradation rate because of acceleration in PDE-mediated hydrolysis, a persistent inhibition of sGC activity, or the presence of both mechanisms. It can be also caused by the depletion of the sGC substrate GTP. This evidence would tend to indicate that the intracellular regulation and control of the NO-induced cGMP response in VSMC is the result of coordinated actions of all the competing factors. The current model could be greatly improved if these factors could be incorporated to analyze the molecular interactions and their biochemical roles, which can help to further elucidate the underlying mechanisms governing the cGMP regulation.

There is still much controversy regarding the existence of cGMP-independent NO mechanisms. Studies on rat basilar arteries suggest that NO causes vasodilation by a cGMP-dependent pathway without affecting the activity of the K_Ca channel (2, 35). In rat middle cerebral arteries, it appears that NO activates K_Ca channels via a cGMP-independent pathway (6, 45). The specific mechanism involved in the cGMP-independent pathway for NO-induced K_Ca channel activation is still unclear and is under investigation. In the present study, the model parameters for the K_Ca channel (Eq. 4) were identified by fitting the channel P_o data (P_o-Ca²⁺ and P_o-V_m relationships, Fig. 5) from Stockand and Sansom (36). Our model shows that sGC desensitization limits the capability of sGC to catalyze cGMP production and higher NO must be handled via an alternative mechanism to the cGMP pathway. The model hypothesizes the coexistence of both cGMP-dependent and cGMP-independent mechanisms. Our model simulations show that a direct NO effect via the K_Ca channel reduces MLC phosphorylation under the conditions of sGC inhibition (Fig. 6), which agrees qualitatively with the experimental study by Onoue and Katusic (29).

The model also suggests that long-term regulation of the expression of K⁺ channels (including K_Ca and K_i via inhibition of NOS) plays an important role in hypertension caused by VSMC membrane depolarization due to chronic endogenous NO deficiency (Fig. 7). Because membrane depolarization induced by elevating [K⁺]_o can be attributed to all types of K⁺ channels expressed on the VSMC membrane, other K⁺ channels [e.g., ATP-sensitive K⁺ (K_ATP) channel] that were not represented in our model could also be potentially affected by long-term NOS inhibition.

In addition, we investigated the Ca²⁺ desensitization effect of the VSMC contractile system by cGMP. The model simulations closely agree with data by Lee et al. (19) for cGMP effects on MLC Ca²⁺-phosphorylation and contractile Ca²⁺-force relationships (Fig. 8, A and B). We distinguish the concept of Ca²⁺ desensitization from Ca²⁺ sensitivity of MLC phosphorylation. As defined in Eq. 5, MLC phosphorylation always has higher sensitivity to Ca²⁺ changes at lower Ca²⁺ levels and cGMP actually increases Ca²⁺ sensitivity _Ca (Fig. 8C). However, Ca²⁺ desensitization as commonly used in the literature means that cGMP causes reduction in MLC phosphorylation at the same level of Ca²⁺.

An integrative cell signaling model like this one must be carefully developed, expanded, and correlated closely with experimental data. Several mechanisms are not currently characterized in our model, but the model will be more valuable if it can take account of the following future objectives.

PKG activation. It has been shown that PKG isoforms (PKG I and -) have different intracellular spatial distributions (11). This indicates that NO/cGMP signals can mobilize PKG to certain regulatory targets that are designated for being phosphorylated by the specific PKG isoforms.

Other membrane targets for cGMP. cGMP can modulate behaviors of other membrane channels and lead to Ca²⁺ reduction. These include deactivation of the L-type Ca²⁺ channel (38), activation of membrane Ca²⁺-ATPase (44), and inhibition of the Ca²⁺-dependent Cl^– channel.

NO/cGMP effects on Ca²⁺ sparks. Ca²⁺ sparks released from the sarcoplasmic reticulum may contribute to global Ca²⁺ concentrations and/or cause membrane hyperpolarization by activating K_Ca channels (18). A recent study shows that NO inhibits the activity of norepinephrine on these sites partly via a cGMP-independent pathway but does not increase spark activity or recruit new so-called frequent discharge sites (30). Other studies, however, suggest that cGMP-mediated VSMC relaxation may occur partially through an increase in the frequency of Ca²⁺ sparks and a subsequent increase in the activity of K_Ca channels (18). Our modeling study would require the knowledge of the mechanisms of Ca²⁺ spark modulations (e.g., frequency modulation and amplitude modulation) by NO/cGMP-related factors.

Despite the above limitations, our model represents an extensible framework for the study of the important NO/cGMP signaling mechanisms. It also can potentially be incorporated into a multiscaled model to study blood vessel behavior under various physiological and pathological conditions (43). Considering the large variety of VSMC types across different tissues and species, NO/cGMP mechanisms are bound to vary and have designs that are specific to cell types. Certain regulatory targets present in one cell type may not be necessarily present in another, or may respond differently to NO signals. Moreover, in most cases, reported experimental results in vitro or on purified molecules were detected under conditions different from those in vivo. Thus the model can serve as a general modeling framework for the study of NO/cGMP effects on the VSMC as a complex cell signaling system. This framework can be tailored to address particular situations and can also be expanded to incorporate newly observed mechanisms.

	GRANTS

TOP ABSTRACT METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
This work was supported by National Institute of Neurological Disorders and Stroke Grant P01-NS-38660.

	ACKNOWLEDGMENTS

 
Present address of J. Yang: Theoretical Biology and Biophysics Group, Theoretical Division, MS-K710, T-10, Los Alamos National Laboratory, Los Alamos, NM 87545 (E-mail: jyang@lanl.gov).

	FOOTNOTES

 

Address for reprint requests and other correspondence: J. W. Clark, Dept. of Electrical and Computer Engineering, MS-366, Rice Univ., Houston, TX 77005 (E-mail: jwc{at}rice.edu)

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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