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Predicted effects of nitric oxide and superoxide on the vasoactivity of the afferent arteriole

Department of Mathematics, Duke University, Durham, North Carolina; and

and

Sorbonne Universités, UPMC Université Paris 06, Université Paris Descartes, Sorbonne Paris Cité, INSERM UMRS 1138, CNRS ERL 8228, Centre de Recherche des Cordeliers, Paris, France

Address for reprint requests and other correspondence: A. Edwards, Centre de Recherche des Cordeliers, 15 rue de l'Ecole de Médecine, 75006 Paris, France (e-mail: [email protected]).

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Published Online:15 Oct 2015https://doi.org/10.1152/ajprenal.00187.2015

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Abstract

We expanded a published mathematical model of an afferent arteriole smooth muscle cell in rat kidney (Edwards A, Layton, AT. Am J Physiol Renal Physiol 306: F34–F48, 2014) to understand how nitric oxide (NO) and superoxide (O₂⁻) modulate the arteriolar diameter and its myogenic response. The present model includes the kinetics of NO and O₂⁻ formation, diffusion, and reaction. Also included are the effects of NO and its second messenger cGMP on cellular Ca²⁺ uptake and efflux, Ca²⁺-activated K⁺ currents, and myosin light chain phosphatase activity. The model considers as well pressure-induced increases in O₂⁻ production, O₂⁻-mediated regulation of L-type Ca²⁺ channel conductance, and increased O₂⁻ production in spontaneous hypertensive rats (SHR). Our results indicate that elevated O₂⁻ production in SHR is sufficient to account for observed differences between normotensive and hypertensive rats in the response of the afferent arteriole to NO synthase inhibition, Tempol, and angiotensin II at baseline perfusion pressures. In vitro, whether the myogenic response is stronger in SHR remains uncertain. Our model predicts that if mechanosensitive cation channels are not modulated by O₂⁻, then fractional changes in diameter induced by pressure elevations should be smaller in SHR than in normotensive rats. Our results also suggest that most NO diffuses out of the smooth muscle cell without being consumed, whereas most O₂⁻ is scavenged, by NO and superoxide dismutase. Moreover, the predicted effects of superoxide on arteriolar constriction are not predominantly due to its scavenging of NO.

to maintain normal kidney function, renal blood flow is tightly regulated over a wide range of perfusion pressure values (7, 33). The two major renal autoregulatory mechanisms are the myogenic response and the tubuloglomerular feedback. The myogenic response elicits a reflex constriction in the preglomerular vasculature in response to a rise in intravascular pressure, thereby generating a compensatory increase in vascular resistance to stabilize the glomerular filtration rate (GFR). The tubuloglomerular feedback mechanism balances GFR with tubular reabsorptive capacity (7, 23, 33). The moygenic response and tubuloglomerular feedback share a common effector in the afferent arteriole.

Nitric oxide (NO) functions as a vasodilator and is an important factor in the maintenance of blood pressure and renal perfusion. Chronic inhibition of NO synthase (NOS) reduces medullary blood flow (6) and, owing to the inhibitory effects of NO on thick ascending limb salt transport, is associated with sodium retention and the development of hypertension (43). NO induces vasodilation through a complex web of signaling pathways; thus a goal of this study is to develop a detailed model of the afferent arteriole smooth muscle cell that incorporates the effects of NO and to use that model to assess the relative contribution of individual effects.

In contrast, superoxide (O₂⁻) acts to reduce medullary blood flow (14) and enhance thick ascending limb sodium reabsorption (19). Elevated production of O₂⁻ has been shown to contribute significantly to the functional alterations of arteries in hypertension (2, 22). Thus the balance between NO and O₂⁻ likely plays a key role in the long-term control of blood pressure. Indeed, enhanced scavenging of NO by O₂⁻, and hence the removal of its buffering influence on vasoconstrictor systems, are believed to contribute significantly to the development of endothelial dysfunction and hypertension (57).

The spontaneously hypertensive rat (SHR) is a widely used animal model of essential hypertension. One interesting characteristic of the SHR is that its autoregulatory plateau (i.e., the range of renal perfusion pressures where single nephron GFR remains approximately constant) is shifted to the right to coincide with its elevated blood pressure range (30). As a result, glomerular capillaries in SHR are better protected against barotrauma, and SHR are substantially less susceptible to renal injury than other models of hypertension (44). The right-shift of the SHR autoregulatory curve has been attributed to the enhanced vasoconstriction exhibited by their afferent arterioles in response to increased perfusion pressure, compared with Wistar-Kyoto (WKY) rats (29).

Studies in SHR indicate that the development of hypertension is associated with increased production of O₂⁻ in vascular tissues (59, 64). Ren et al. (46) demonstrated in microdissected afferent arterioles that elevated transmural pressure increases O₂⁻ production, and this increase is particularly pronounced in SHR. These investigators postulated that the enhanced myogenic response of SHR can be attributed, in part, to the activation of NADPH oxidase (NOX) and the increased generation of O₂⁻. The principal goal of this study is to use a detailed mathematical model of the afferent arteriole smooth muscle cell to investigate how altering the balance between NO and O₂⁻ affects arteriolar vasoconstriction.

MATHEMATICAL MODEL

We expanded a published model of an afferent arteriole smooth muscle cell in rat kidney (11) to include the effects of NO and O₂⁻. Figure 1 displays the main signaling pathways for which our model accounts. The contractile force exerted on the cell depends on the fraction of myosin light chains (MLC) that is phosphorylated. The model represents the effects of NO and O₂⁻ on plasma membrane (PMCA) and sarco/endoplasmic reticulum (SERCA) Ca²⁺ pumps, Na⁺-K⁺-ATPase, Ca²⁺-gated K⁺ channels, and L-type Ca²⁺ channels and on the activation of myosin light chain phosphatase (MLCP).

**Fig. 1.**Schematic diagram of the signaling pathways in an afferent arteriole smooth muscle cell. Not shown are background channels, inward- and delayed-rectifier K⁺ channels, and buffering reactions. The contractile force depends on the fraction of myosin light chains (MLC) that are phosphorylated. The model represents the effects of nitric oxide (NO) and O₂⁻ on Ca²⁺-activated K⁺ (K_Ca) and L-type Ca²⁺ channels, plasma membrane (PMCA) and sarco/endoplasmic reticulum (SERCA) Ca²⁺ pumps, Na⁺-K⁺-ATPase, and myosin light chain phosphatase (MLCP) activation. MLCK, myosin light chain kinase; NCX, Na⁺/Ca²⁺ exchanger. See text for meaning of other symbols.

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Ion and Charge Conservation Equations

The cytosolic concentrations of K⁺, Na⁺, Cl⁻, and Ca²⁺ as a function of time are determined by considering the sum of their respective fluxes into the cytosol and integrating the following differential equations:

\frac{d {[K]}_{cyt}}{d t} = - \frac{(I_{K,b} + I_{K,ir} + I_{K,v} + I_{K,Ca} - 2 I_{NaK})}{F \cdot {vol}_{cyt}}

(1)

\frac{d {[Na]}_{cyt}}{d t} = - \frac{(I_{Na,b} + I_{Na,Pres} + 3 I_{NaK} + 3 I_{NCX})}{F \cdot {vol}_{cyt}} + \frac{J_{NaCl}}{{vol}_{cyt}}

(2)

\frac{d {[Cl]}_{cyt}}{d t} = + \frac{(I_{Cl,b} + I_{Cl, Ca})}{F \cdot {vol}_{cyt}} + \frac{J_{NaCl}}{{vol}_{cyt}}

(3)

\frac{d {[Ca]}_{cyt}}{d t} = - \frac{(I_{Ca,b} + I_{Ca,Pres} + I_{PMCA} + I_{Ca,L} - 2 I_{NCX} + I_{SERCA} - I_{R y R} - I_{IP3R})}{2 F \cdot {vol}_{cyt,Ca}} + R_{CaM}^{cyt} + R_{Bf}^{cyt}

(4)

where F is Faraday's constant and vol_cyt and vol_cyt,Ca denote the total volume of the cytosol and the volume that is available to Ca²⁺, respectively. The convention adopted in this study is that exit of positive charge from the cell is a positive current. Conversely, exit of negative charge is a negative current. The following currents are not affected by NO and O₂⁻ and can be found in Ref. 11: the background current of ion i (I_i,b), potassium currents across inward-rectifier and delayed-rectifier channels (I_K,ir and I_K,v), the current across Na⁺/Ca²⁺ exchangers (I_NCX), the flux across NaCl cotransporters (J_NaCl), the chloride current across Ca²⁺-activated Cl⁻ channels (I_Cl,Ca), as well as calcium currents mediated by ryanodine and inositol-trisphosphate receptors (I_RyR and I_IP3R). Other currents are described below. Calcium buffering by calmodulin (R_CaM^cyt) and by other cytosolic buffers (R_Bf^cyt) is described in Ref. 11.

In the sarcoplasmic reticulum (SR), we have:

\frac{d {[Ca]}_{SR}}{d t} = + \frac{(I_{SERCA} - I_{RyR} - I_{IP3R})}{2 F \cdot {vol}_{SR}} + R_{Calseq}^{S R}

(5)

where vol_SR denotes the volume of the SR. The reaction term R_Calseq^SR accounts for the buffering of Ca²⁺ by calsequestrin and is described in Ref. 11.

The transmembrane potential V_m is determined by the net sum of the currents flowing across the plasma membrane (11).

Kinetics of Formation of NO and O₂⁻

Nitric oxide.

The kinetics of NO and O₂⁻ formation, diffusion, and reaction are modeled as in our previous study of NO-induced vasodilation of descending vasa recta (10). We assume that the only source of NO is the endothelium. Once generated, endothelial NO diffuses into the smooth muscle cell where it is partly consumed by several scavengers (e.g., O₂⁻, O₂, and thiols) while the rest diffuses across the abluminal cell membrane into the interstitium. Conservation of NO is given by:

{vol}_{cyt} \frac{d {[NO]}_{cyt}}{d t} = \frac{A}{2} J_{NO}^{endo} - \frac{A}{2} P_{NO} {[NO]}_{cyt} - {vol}_{cyt} k_{{NO,O}_{2}^{-}} {[NO]}_{cyt} {[O_{2}^{-}]}_{cyt} - {vol}_{cyt} k_{NO,cons} {[NO]}_{cyt}

(6)

where A (taken as 5.5 × 10⁻⁶ cm²) is the total membrane surface area; J_NO^endo, which denotes the NO flux from the endothelium, is fixed at 6.4 × 10⁻⁶ μmol·cm⁻²·s⁻¹ so that the basal concentration of NO is on the order of 100 nM, as measured in tubular fluid in vivo (36). The second term on the right-hand side represents NO diffusion towards the interstitium: P_NO (estimated as 0.03 cm/s) is the cell membrane permeability to NO, and peritubular [NO] is assumed to be much smaller than [NO]_cyt. The third term describes the reaction between NO and O₂⁻ that yields perixynitrite (ONOO⁻):

{NO+O}_{2}^{-} \overset{k_{{NO,O}_{2}^{-}}}{\to} ONO O^{-}

(7)

where the rate constant $k_{NO, O_{2}^{-}}$ is set to 6.7 × 10³ μM⁻¹·s⁻¹ (25). Scavenging of NO by species other than O₂⁻ is lumped into a first-order process with a rate constant denoted by k_NO,cons, taken as 0.01 s⁻¹.

Superoxide.

Whereas NO is produced by the endothelium, O₂⁻ is generated within the smooth muscle cell. It then either exits the cell via diffusion or reacts with scavengers. Changes in cytosolic [O₂⁻] are given by:

{vol}_{cyt} \frac{d {[O_{2}^{-}]}_{cyt}}{d t} = {vol}_{cyt} G_{O_{2}^{-}} - \frac{A}{2} P_{O_{2}^{-}} {[O_{2}^{-}]}_{cyt} (\frac{ζ}{1 - e^{- ζ}}) - {vol}_{cyt} k_{NO, O_{2}^{-}} {[NO]}_{cyt} {[O_{2}^{-}]}_{cyt} - {vol}_{cyt} k_{SOD, O_{2}^{-}} {[SOD]}_{cyt} {[O_{2}^{-}]}_{cyt}

(8)

where $P_{O_{2}^{-}}$ (estimated as 0.06 cm/s) is the cell membrane permeability to O₂⁻; because O₂⁻ is charged, the diffusion term depends on the nondimensional transmembrane electric potential difference, given by ζ = −FV_m/RT. The volumetric generation rate of O₂⁻ ( $G_{O_{2}^{-}}$ ) is taken to vary with the luminal pressure P (in mmHg) when the latter is >100 mmHg:

G_{O_{2}^{-}} = {\begin{matrix} G_{O_{2}^{-}}^{*} α_{O_{2}^{-}}, & P < 100 mmHg, \\ G_{O_{2}^{-}}^{*} α_{O_{2}^{-}} (1 + β_{O_{2}^{-}} (\frac{P - 100}{40})), & P \geq 100 mmHg \end{matrix}

(9)

Ren et al. (46) reported that pressure-induced increases in O₂⁻ were fourfold greater in SHR afferent arterioles compared with WKY afferent arterioles (48 vs. 12%, as perfusion pressure increased from 60 to 140 mmHg). Moreover, the basal rate of O₂⁻ synthesis is higher in hypertensive rats (37). Given these observations, the parameters $α_{O_{2}^{-}}$ and $β_{O_{2}^{-}}$ are set to 1.0 and 0.12, respectively, for normotensive rats, and to 2.5 and 0.48, respectively, for SHR. The reference generation rate $G_{O_{2}^{-}}^{*}$ is taken as 14.4 μM/s.

The two sinks for O₂⁻ include its reaction with NO to form ONOO⁻ (Eq. 7) and the reaction catalyzed by superoxide dismutase (SOD) to form H₂O₂:

{2O}_{2}^{-} + {2H}^{+} + {SOD}_{.} \overset{k_{SOD, O_{2}^{-}}}{\to} H_{2} O_{2} + O_{2} + SOD

(10)

The reaction rate is given by the product $k_{SOD, O_{2}^{-}}$ [SOD]_cyt[O₂⁻]_cyt, where [SOD]_cyt and $k_{SOD, O_{2}^{-}}$ are, respectively, set to 1 μM and 1.6 × 10³ μM⁻¹·s⁻¹. Owing to insufficient quantitative data, the effects of ONOO⁻ and H₂O₂ on afferent arteriole vasomotor responses are not considered in this model.

Soluble Guanylate Cyclase Activation and cGMP Generation

Activation of soluble guanylate cyclase by NO.

The activation of soluble guanylate cyclase (sGC) by NO is modeled following the approach of Yang et al. (61). First, in a reversible reaction, NO binds to basal sGC (denoted sGC_basal) to yield an intermediate six-coordinate ferrous nitrosyl heme complex (denoted sGC_6c). Then, sGC_6c is converted to a fully activated five-coordinate complex (denoted sGC_5c). This second step is believed to occur via two single-step irreversible processes, one NO dependent and the other NO independent. Finally, the deactivation of sGC_5c is represented as a first-order, irreversible reaction. The kinetic equations are expressed as:

\frac{d [{sGC}_{basal}]}{d t} = - k_{1}^{sGC} [{sGC}_{basal}] [NO] + k_{- 1}^{sGC} [{sGC}_{6c}] + k_{4}^{sGC} [{sGC}_{5c}]

(11)

\frac{d [{sGC}_{6c}]}{d t} = + k_{1}^{sGC} [{sGC}_{basal}] [NO] - k_{- 1}^{sGC} [{sGC}_{6c}] - k_{2}^{sGC} [{sGC}_{6c}] - k_{3}^{sGC} [{sGC}_{6c}] [NO]

(12)

For simplicity, the subscript “cyt” is omitted in the conservation equations of species other than the main ions, NO, and O₂⁻. The concentration of sGC_5c can be determined from the conservation equation:

[{sGC}_{basal}] + [{sGC}_{6c}] + [{sGC}_{5c}] = {[sGC]}_{tot}

(13)

where the total sGC concentration ([sGC]_tot) is fixed, since we assume that cytosolic volume remains constant. The reaction rates (k_i^sGC) in Eqs. 11 and 12 are taken to be constant, with two exceptions. The conversion of sGC_5c to basal sGC is assumed to be mediated by cGMP. Thus

k_{4}^{sGC} = K_{4}^{sGC} [cGMP]

(14)

where K₄^sGC = 0.098 μM⁻¹·s⁻¹. As discussed below, the production rate of cGMP increases with [sGC_5c], which implies a negative feedback loop.

Oxidative stress has been shown to alter the redox state of the prosthethic heme on sGC, such that the latter becomes unresponsive to NO (53). To account for the inhibitory effects of oxidative stress on NO-mediated activation of sGC, we assume that the rate k₁^sGC is reduced by a factor of 10 when [O₂⁻]_cyt is above 10 nM (i.e., twice its base-case value).

Generation and degradation of cGMP.

The synthesis of cGMP is represented as in Ref. 61. GTP is converted to cGMP at a rate that depends on sGC_5c levels. Cyclic nucleotide phosphodiesterases (PDE) then hydrolyze cGMP into GMP. That degradation pathway is modeled using Michaelis-Menten kinetics, with the maximum rate taken to be proportional to [cGMP], to account for PDE activation by cGMP. Taken together, the kinetics of cGMP formation is given by:

\frac{d [cGMP]}{d t} = V_{max, sGC} [{sGC}_{5c}] - \frac{k_{P D E} {[cGMP]}^{2}}{K_{M, PDE} + [cGMP]}

(15)

Mechanosensitive Channels

As discussed below, the identity of the mechanosensitive cation channels activated by wall stress that are expressed by the afferent arteriole remains to be fully elucidated. We assume that these channels predominantly carry Na⁺ and that they are somewhat permeable to Ca²⁺ but not to other ions. The corresponding currents are calculated as:

I_{Na, Pres} = G_{Na, Pres} (V_{m} - E_{Na})

(16)

I_{Ca, Pres} = G_{Ca, Pres} (V_{m} - E_{Ca})

(17)

where the conductance G_i,Pres to ion i (i = Na⁺, Ca²⁺) is taken to depend on the luminal pressure P as follows:

G_{i, Pres} = {\begin{matrix} G_{i, Pres}^{0} (1 + α_{Pres} (\frac{P}{P_{1}} - 1)), & P < P_{2} \\ G_{i, Pres}^{0} (1 + α_{Pres} (\frac{P}{P_{1}} - 1) (\frac{P}{P_{2}})), & P \geq P_{2} \end{matrix}

(18)

with P₁ ≡ 100 mmHg. P₂ is set to 60 mmHg, consistent with the finding that the myogenic response becomes ineffective at sufficiently low luminal pressure (7). The base-case value of α_Pres, which characterizes the pressure sensitivity of the mechanosensitive channels, is taken as 1.75.

NO and cGMP Effects on K⁺ Transport Pathways

As in Ref. 61, the model considers that NO and cGMP can separately shift the voltage dependence of the open probability of Ca²⁺-activated K⁺ (K_Ca) channels toward more negative potentials. The current flowing through these channels is calculated as:

I_{K,Ca} = G_{KCa} P_{KCa} (V_{m} - E_{K})

(19)

P_{KCa} = 0.65 P_{f} + 0.35 P_{s}

(20)

\frac{d P_{f}}{dt} = \frac{{\bar{P}}_{KCa} - P_{f}}{τ_{P f}}

(21)

\frac{d P_{s}}{dt} = \frac{{\bar{P}}_{KCa} - P_{s}}{τ_{P s}}

(22)

{\bar{P}}_{KCa} = \frac{1}{1 + \exp (- \frac{V_{m} - V_{KCa}}{k_{KCa}})}

(23)

V_{KCa} = V_{Ca}^{KCa} \log_{10} ({[Ca]}_{cyt}) - V_{NO}^{KCa} (\frac{[NO]}{K_{M, NO}^{KCa} + [NO]}) - V_{cGMP}^{KCa} (\frac{[cGMP]}{K_{M, cGMP}^{KCa} + [cGMP]})

(24)

where P_f and P_s denote the fast and slow components of the channel activation process, respectively, and τ_Pf and τ_Ps denote the corresponding time constants. The steady-state open probability of the channel is given by P̄_KCa. Parameter values are given in Table 5 in Ref. 10.

O₂⁻ Effects on K⁺ and Na⁺ Transport Pathways

The current across Na⁺-K⁺-ATPase pumps is determined as:

I_{NaK} = I_{NaK, max} {(\frac{{[K]}_{out}}{{[K]}_{out} + K_{M, NaK}^{K}})}^{2} {(\frac{{[Na]}_{cyt}}{{[Na]}_{cyt} + K_{M, NaK}^{Na}})}^{3}

(25)

The pump affinities are taken from Refs. 39 and 63. To account for the observed inhibitory effects of O₂⁻ on Na⁺-K⁺-ATPase activity (13), we assume that O₂⁻ modulates the maximum current as:

I_{NaK,max} = I_{NaK, max}^{*} [1 - \frac{{[O_{2}^{-}]}_{cyt}}{{[O_{2}^{-}]}_{cyt} + K_{M, O_{2}^{-}}^{NaK}}]

(26)

where $K_{M, O_{2}^{-}}^{NaK}$ is set to 1.6 μM.

Effects of cGMP, NO, and O₂⁻ on Ca²⁺ Transport Pathways

cGMP-dependent PKG has been found to stimulate PMCA pumps (62). The current through PMCA is expressed as:

I_{PMCA} = I_{PMCA, max} (\frac{{[Ca]}_{cyt}}{K_{M, P M C A}^{Ca} + {[Ca]}_{cyt}})

(27)

where

I_{PMCA,max} = I_{PMCA, max}^{*} (1 + 0.1 \frac{[cGMP]}{K_{M, cGMP}^{PMCA} + [cGMP]})

(28)

The maximum current in the absence of cGMP, I_{PMCA, max}^*, is taken as 4.5 pA. The affinities for Ca²⁺ (K_M,PMCA^Ca) and for cGMP (K_M,cGMP^PMCA) are taken as 0.170 μM (18) and 1.0 μM (10), respectively. To obtain a better fit with afferent arteriole data, we assume a smaller dependence of the PMCA current on cGMP levels than in our descending vasa recta model (10).

At sufficiently high concentrations (≥0.1 μM), cGMP has been shown to activate SERCA pumps via phosphorylation of phospholamban (5), whereas O₂⁻ inhibits SERCA (34). The SERCA current is determined as:

I_{SERCA} = I_{SERCA, max} (\frac{{({[Ca]}_{cyt})}^{2}}{{({[Ca]}_{cyt})}^{2} + {(K_{mf})}^{2}})

(29)

where

K_{mf} = K_{mf}^{*} (1 - \frac{1}{2} \frac{[cGMP]}{K_{M,cGMP}^{SERCA} + [cGMP]})

(30)

where K_M,cGMP^SERCA is set to 0.50 μM (5) and K_mf^* is taken as 0.59 μM (10). The inhibitory effects of O₂⁻ are captured in the maximum SERCA current term:

I_{SERCA,max} = I_{SERCA,max}^{*} (1 - \frac{{({[O_{2}^{-}]}_{cyt})}^{0.5}}{{(K_{M, O_{2}^{-}}^{SERCA})}^{0.5} + {({[O_{2}^{-}]}_{cyt})}^{0.5}})

(31)

The maximum current, $I_{SERCA, \max}^{*}$ , is taken as 11.8 pA and $K_{M, O_{2}^{-}}^{SERCA}$ as 20 μM (12).

The current across L-type Ca²⁺ channels (Ca_V1.2) is given by:

I_{Ca, L} = G_{Ca,L} p_{o} (V_{m} - E_{Ca})

(32)

where G_Ca,L represents the maximum conductance of the channel, and p_o, which denotes the open state of the channel, is determined using the model of Faber et al. (15). Model equations can be found in Ref. 11. Several studies have shown that reactive oxygen species (ROS) stimulate L-type Ca²⁺ channel activity (1, 24). Thus we assume that the maximum conductance of the channel is dependent on cytosolic [O₂⁻] above a certain threshold:

G_{Ca, L} = {\begin{matrix} G_{C a, L}^{*}, & {[O_{2}^{-}]}_{cyt} < {[O_{2}^{-}]}_{100} \\ G_{C a, L}^{*} (1 + β_{C a, L} (\frac{{[O_{2}^{-}]}_{cyt}}{{[O_{2}^{-}]}_{100}} - 1)) & {[O_{2}^{-}]}_{cyt} \geq {[O_{2}^{-}]}_{100} \end{matrix}

(33)

where [O₂⁻]₁₀₀ (taken as 5.0 nM) corresponds to the predicted cytosolic concentration of superoxide in normotensive rats at a luminal pressure of 100 mmHg. Based on the data of Smirnov et al. (52) and this model, the reference maximum conductance of the L-type channel (G_Ca,L^*) is estimated as 0.5 nS/pF or 2.75 nS. The base-case value of β_Ca,L is taken as 0.10.

MLCK- and MLCP-Dependent Phosphorylation of Myosin

The fraction of myosin cross bridges that are phosphorylated determines the contractile force. Following the approach of Yang et al. (61), we consider four species: free cross bridges (Myo), phosphorylated cross bridges (MyoP), attached phosphorylated, cycling cross bridges (AMyoP), and attached dephosphorylated, noncycling cross bridges (AMyo). The concentration of these MLC species is calculated as:

\frac{d [Myo]}{d t} = - k_{1}^{Myo} [Myo] + k_{2}^{Myo} [MyoP] + k_{7}^{Myo} [AMyo]

(34)

\frac{d [MyoP]}{d t} = + k_{1}^{Myo} [Myo] - k_{2}^{Myo} [MyoP] - k_{3}^{Myo} [MyoP] + k_{4}^{Myo} [AMyoP]

(35)

\frac{d [AMyoP]}{d t} = + k_{3}^{Myo} [MyoP] - k_{4}^{Myo} [AMyoP] - k_{5}^{Myo} [AMyoP] + k_{6}^{Myo} [AMyo]

(36)

[Myo] + [MyoP] + [AMyoP] + [AMyo] = {[Myo]}_{tot}

(37)

The rate constants k₃^Myo, k₄^Myo, and k₇^Myo are fixed (60). The rate constants k₁^Myo and k₆^Myo represent the activity of MLCK, and are taken to be proportional to the fraction of the fully activated form of the enzyme (the one that is bound to calmodulin and 4 Ca²⁺ ions):

k_{1}^{Myo} = k_{6}^{Myo} = k_{MLCK}^{Myo} \frac{[MLCK . CaM . {Ca}_{4}]}{{[MLCK]}_{t o t}}

(38)

where k_MLCK^Myo has a fixed value of 0.2135 s⁻¹.

The rate constants k₂^Myo and k₅^Myo represent the activity of MLCP. cGMP has been shown to indirectly activate MLCP, which may underlie the observed cGMP-induced Ca²⁺ desensitization in vasodilation (35). The cGMP effects are accounted for as:

k_{2}^{Myo} = k_{5}^{Myo} = k_{MLCP}^{Myo} f_{MLCP}^{a c t} (1 + 0.1 \frac{{[cGMP]}^{2}}{{(K_{M, cGMP}^{MLCP})}^{2} + {[cGMP]}^{2}})

(39)

where k_MLCP^Myo is a constant (set to 0.70 s⁻¹), f_MLCP^act is the fraction of active (dephosphorylated) MLCP, and K_M,cGMP^MLCP is taken as 5.5 μM (61). MLCP is also negatively regulated by Rho kinase (RhoK). The fraction f_MLCP^act is determined following the approach of Mbikou et al. (41):

\frac{d f_{MLCP}^{act}}{d t} = k_{- P} - (k_{+ P} + k_{- P}) f_{MLCP}^{act}

(40)

The dephosphorylation rate k_−P is taken to remain constant (0.2 s⁻¹), whereas the phosphorylation rate k_+P varies with the fraction of active RhoK (f_RhoK^act) as:

k_{+ P} = \frac{k_{catP}}{K_{MP}} f_{RhoK}^{act} {[RhoK]}_{tot}

(41)

where k_catP (2.5 s⁻¹) and K_MP (2.5 μM) are the corresponding enzyme-substrate breakdown constant and Michaelis constant, respectively, and [RhoK]_tot is the total concentration of RhoK (fixed at 2.0 μM) (41). In turn, the fraction of active RhoK is given by:

\frac{d f_{RhoK}^{act}}{d t} = k_{+ R K} - (k_{+ R K} + k_{- R K}) f_{RhoK}^{act}

(42)

where the “off” rate constant k_−RK is set to 0.1 s⁻¹. The activation of RhoK is Ca²⁺ dependent and the “on” rate constant k_+RK is calculated as:

k_{+ RK} = γ_{RK} {[Ca]}_{cyt}

(43)

where γ_RK is taken as 0.3 μM·s⁻¹ in normotensive rats (41). There is evidence that oxidative stress activates RhoA/RhoK signaling in vascular smooth muscle cells (3, 31, 32), but the underlying signaling cascades remain to be fully characterized. To incorporate these ROS-dependent effects in the present study, we simply assume that γ_RK is multiplied by a factor of 1.5 when [O₂⁻]_cyt is >10 nM.

Vessel Mechanics

Variations in the number of cross bridges induce variations in the contractile force, which in turn modulate the arteriolar diameter. To simulate vasomotion, the model represents the vascular wall tension as the sum of a passive component, T_pass, and an active myogenic component, given by the product of the maximal active tension that is generated at a given vessel circumference, T_act^max, and the fraction of MLC that is phosphorylated. Thus, the total tension in the wall T_wall is given by:

T_{wall} = T_{pass} + (\frac{[MyoP] + [AMyoP]}{{[Myo]}_{tot}}) T_{act}^{max}

(44)

The passive tension is a function of the afferent arteriole diameter D:

T_{pass} = C_{pass}^{″} + C_{pass} \exp (C_{pass}^{'} (\frac{D}{D_{0}} - 1))

(45)

where D₀ denotes the reference arteriolar diameter. The maximal active tension is given by:

T_{act}^{max} = C_{act} \exp (- {(\frac{D / D_{0} - C_{act}^{′}}{C_{act}^{″}})}^{2})

(46)

The vessel diameter changes according to the difference between the tension T_pres resulting from intravascular pressure p, given by T_pres = PD/2, and the tension generated within the wall, T_wall:

\frac{d D}{d t} = \frac{1}{τ_{d}} (T_{pres} - T_{wall})

(47)

where τ_d is a time constant. Parameter values for the muscle mechanics model are given in Table 1. Parameter values related to NO, cGMP, and O₂⁻ are given above. Other parameter values can be found in Refs. 10 and 11, with the following modifications: G_ClCa = 0.48 nS, K_ClCa = 100 nM, and υ_RyR = 14 s⁻¹.

**Table 1. Smooth muscle cell mechanical parameters**
Parameter	Value	Unit
C_pass	1,693.07	dyn/cm
C′_pass	6.0	Dimensionless
C″_pass	−2,250.0	dyn/cm
C_act	15,287.7	dyn/cm
C′_act	1.0	Dimensionless
C″_act	0.472965	Dimensionless
D₀	24.0	μm
τ_d	2.0	s

MODEL RESULTS

We examined the response of the rat afferent arteriole to NO, O₂⁻, angiotensin II (ANG II), and perfusion pressure in normotensive and SHR. Parameter values that differ between these strains are summarized in Table 2 and discussed below.

**Table 2. Assumed differences between rat strains**
Parameter	SD	SHR
$G_{O_{2}^{-}}$ , μM/s	14.4	36.0
k₁^sGC, μM/s	2 × 10³	2 × 10²
γ_RK, μM/s	0.30	0.45

$G_{O_{2}^{-}}$ , O₂⁻ production rate at 100 mmHg; k₁^sGC, forward rate constant of the reaction between nitric oxide and basal soluble guanylate cyclase (Eq. 11); γ_RK, Ca²⁺ dependence of RhoK activation (Eq. 43); SD, Sprague-Dawley rats; SHR, spontaneously hypertensive rats.

Effects of Superoxide on the Basal Diameter of the Afferent Arteriole

When the luminal pressure is set to its base-case value of 100 mmHg, the model predicts spontaneous vasomotion. In normotensive rats, the cytosolic concentration of Ca²⁺ (denoted [Ca²⁺]_cyt below) oscillates between 180 and 320 nM, with a frequency of 145 mHz. The cytosolic concentration of NO ([NO]_cyt) and O₂⁻ ([O₂⁻]_cyt) equals 152 and 4.9 nM, respectively. The average afferent arteriole luminal diameter is 20.5 μm, with an oscillation amplitude of ∼1.9 μm.

The superoxide production rate in hypertensive rats is taken to be 2.5 times greater than that of normotensive rats (37). Thus, in SHR, the predicted values of [NO]_cyt and [O₂⁻]_cyt are 100 and 14 nM, respectively. Cytosolic Ca²⁺ oscillations are almost negligible: [Ca²⁺]_cyt varies between 272 and 274 nM. The arteriolar luminal diameter is 17.1 μm, that is, significantly lower than in normotensive rats, as observed experimentally (20, 46, 51).

In this model, elevated superoxide levels modulate Ca²⁺ signaling in several ways: by raising the conductance of L-type Ca²⁺ channels (G_Ca,L), by rendering basal sGC less responsive to NO, by scavenging NO, and by activating RhoK, which in turn inactivates MLCP. To determine the relative contribution of these mechanisms to the reduced diameter in SHR, we abolished each effect separately. When G_Ca,L was maintained at its reference (normotensive) value, the afferent arteriolar diameter in SHR was computed as 18.4 μm; when the rate of NO scavenging by O₂⁻ was kept at its normotensive value, the diameter was 17.6 μm; without ROS-induced sCG desensitization to NO, the diameter was 17.4 μm; lastly, without ROS-mediated activation of RhoK, it was 18.1 μm. Thus ROS-induced modulation of G_Ca,L is predicted to be the dominant mechanism by which superoxide affects vasoconstriction.

Effects of NO Synthase Inhibition and Tempol

NO and its second messenger cGMP favor vasodilation by maintaining low levels of Ca²⁺ and MLC phosphorylation. They do so by stimulating Ca²⁺ uptake from the cytosol into the SR (via SERCA pumps) and Ca²⁺ efflux into the extracellular space (via PMCA pumps), activating K_Ca channels to induce hyperpolarization, and enhancing MLCP activity (Fig. 1). Our model predicts that when NO synthesis is fully inhibited, spontaneous vasomotion is abolished and the luminal diameter decreases to 16.0 μm in normotensive rats, a 22% reduction that is close to the measured value (18%) in Sprague-Dawley (SD) rats treated with N-nitro-l-arginine (27). In SHR, NOS inhibition is also predicted to reduce the diameter from 17.1 to 14.5 μm; this 15% decrease is identical to in vitro findings (26).

Experimental studies have found that the SOD mimetic Tempol (100 μM) has no significant effect on the afferent arteriole of normotensive rats under basal conditions (26, 50). We simulated the addition of Tempol by increasing the concentration of SOD from 1 to 100 μM. As observed experimentally, at the baseline pressure of 100 mmHg, the model predicts that Tempol has little impact on the arteriolar diameter in normotensive rats, because O₂⁻ levels are too low to significantly affect NO concentrations and Ca²⁺ signaling pathways. In SHR, Tempol is predicted to increase the arteriolar diameter (by 25%), as seen in many studies (47, 48, 26), albeit with exceptions (46).

We also performed simulations in which we eliminated the downstream effects of NO and cGMP one by one, so as to assess the contribution of each mechanism to the maintenance of basal tone in normotensive rats. Results are summarized in Fig. 2. Abolishing cGMP-induced stimulation of MLCP does not affect [Ca²⁺]_cyt but reduces the basal arteriolar diameter by 0.6 μm. Abolishing the direct and indirect effects of NO on K_Ca channels slightly depolarizes the smooth muscle cell, raises the average [Ca²⁺]_cyt by ≈15 nM, and reduces the arteriolar diameter by 1.7 μm. Eliminating cGMP-induced activation of PMCA has a larger impact: it lowers the diameter by 2.3 μm. Lastly, in the absence of cGMP-mediated effects on SERCA, calcium oscillations vanish and [Ca²⁺]_cyt remains constant. Even though the rate of Ca²⁺ uptake into the SR is then reduced compared with its average base-case value, [Ca²⁺]_cyt is slightly lower than its average base-case value, owing to nonlinear effects (i.e., the interplay between Ca²⁺ uptake into/release from the SR). As a result, the predicted arteriolar diameter is 0.4 μm higher than the average diameter in the base case.

**Fig. 2.**Contributions of individual vasodilatory pathways in the afferent arteriole of Sprague-Dawley (SD) rats, assuming a luminal pressure of 100 mmHg. Red bars indicate the range of cytosolic Ca²⁺ concentration (A) and vessel diameter (B) in spontaneous vasomotion. MLCP, without cGMP modulation of MLC phosphatase activity; K-NO, without NO modulation of K_Ca current; K-cGMP, without cGMP modulation of K_Ca current; both, without NO and cGMP modulation of K_Ca current; PMCA, without cGMP modulation of PMCA activity; SERCA, without cGMP modulation of SERCA activity; none, in the absence of NO and cGMP.

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Effects of ANG II

We simulated the effects of ANG II by increasing the generation rate of IP₃ and O₂⁻ twofold and lowering the conductance of K⁺ channels by 60% (12, 54). With these assumptions, the model predicts that ANG II abolishes vasomotion and reduces the arteriolar diameter to 13.2 μm in normotensive rats (a 36% decrease) and 12.4 μm in SHR. Measured values of the fractional diameter decrease induced by ANG II range from 20 to 60% in SD, depending on the concentration of ANG II (27, 50, 55).

In vitro, NOS inhibition enhances ANG II-induced vasoconstriction. Ikenaga et al. (27) observed that in SD juxtamedullary arterioles 10 nM ANG II reduced the arteriolar lumen by 35%; in the presence of N-nitro-l-arginine, the same concentration of ANG II lowered the diameter by 50%. Our model predicts similar trends: in the presence of ANG II and with full NOS inhibition, the predicted arteriolar diameter is 12.4 μm in normotensive rats, a 40% reduction from the base-case value (Fig. 3).

**Fig. 3.**Effects of NO, O₂⁻, and angiotensin (ANG) II on the afferent arteriole diameter of SD and spontaneously hypertensive rats (SHR), under different conditions: NOS-null, full nitric oxide synthase (NOS) inhibition; Tempol, 100 μm; ANG II, addition of ANG II simulated with a 2-fold increase in IP₃ and O₂⁻ production, and 60% inhibition of K⁺ channels; ANG II-a, ANG II and full NOS inhibition; ANG II-b, ANG II with NO donor (900 nM NO). Red bars indicate the diameter range in spontaneous vasomotion.

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Conversely, NO donors have been shown to reverse ANG II-induced vasoconstriction. At 900 nM, NO was found to abolish almost entirely the effects of ANG II in SD rats (55). In accordance with these experimental results, our model predicts that in the presence of ANG II and a NO donor yielding [NO]_cyt = 900 nM, the arteriolar diameter is 18.0 μm in normotensive rats, that is, 88% of its baseline value (Fig. 3).

Effects of Luminal Pressure Increases

Increases in luminal pressure are thought to elicit vasoconstriction via a signaling cascade that involves the opening of nonselective, mechanosensitive cation channels and the subsequent depolarization of the smooth muscle cell, followed by Ca²⁺ entry through voltage-gated channels (42). Several theoretical studies, including our previous model of the myogenic response in the rat afferent arteriole, have demonstrated the plausibility of this mechanism (4, 11). The present model, in addition, accounts for recent evidence that ROS production increases in response to elevated pressure (46, 56). In this model, the O₂⁻ generation rate is taken to increase linearly with pressure beyond 100 mmHg (Eq. 9), with a proportionality constant that differs between normotensive and hypertensive rats, based on experimental evidence (46). Since the conductance of L-type Ca²⁺ channels (G_Ca,L) is assumed to vary with [O₂⁻]_cyt, as observed experimentally (1, 24), changes in perfusion pressure, via their effect on the rate of O₂⁻ synthesis, modulate G_Ca,L.

The pressure dependence of the mechanosensitive channel conductance and the [O₂⁻]_cyt dependence of the L-type channel conductance remain to be fully described. In our model, they are respectively characterized by the proportionality constants α_PRES (Eq. 18) and β_Ca,L (Eq. 33), which are set to 1.75 and 0.10, respectively, in the base case. Shown in Figs. 4 and 5 are the predicted effects of an increase in luminal pressure on the arteriolar diameter for different values of α_PRES and β_Ca,L.

**Fig. 4.**Predicted effects of varying the pressure-dependence of mechanosensitive channels (α_PRES) on the myogenic response of SD and SHR rats. A: afferent arteriolar diameter at a luminal pressure of 100, 120, 140, and 160 mmHg, for 3 values of α_PRES (baseline α_PRES = 1.75). B: fractional change in diameter, relative to that at 100 mmHg.

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**Fig. 5.**Predicted effects of varying the [O₂⁻]_cyt dependence of the L-type channel conductance (β_Ca,L) on the myogenic response of SHR. A: afferent arteriolar diameter at a luminal pressure of 100, 120, 140, and 160 mmHg, for 3 values of β_Ca,L (baseline β_Ca,L = 0.10). B: fractional change in diameter, relative to that at 100 mmHg. The myogenic response in normotensive rats is insensitive to changes in β_Ca,L.

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With base-case parameter values, the model predicts that a step change in pressure from 100 to 160 mmHg lowers the arteriolar diameter by 35% (from 20.5 to 13.3 μm) in normotensive rats, and by 25% (from 17.1 to 12.8 μm) in SHR. In relative terms, pressure-induced changes in diameter are smaller in SHR because the fraction of MLC that is phosphorylated (which determines the vascular radius) becomes less sensitive to changes in [Ca²⁺]_cyt as the latter is increased. In normotensive rats, the proportion of MLC that is phosphorylated increases from an average value of 0.31 at 100 mmHg to 0.78 at 160 mmHg; in SHR, it increases from 0.42 at 100 mmHg to 0.85 at 160 mmHg. Thus the up-step in pressure elicits a proportionally smaller change in radius in the latter strain.

As depicted in Fig. 4, the amplitude of the myogenic response predicted by the model depends significantly on the value of α_PRES. If α_PRES is increased 2-fold to 3.50, raising the luminal pressure from 100 to 160 mmHg lowers the arteriolar diameter by 39% in normotensive rats and 27% in SHR. Conversely, if α_PRES is set to half its base-case value, the corresponding diameter reductions are 28 and 22%, respectively.

In normotensive rats, pressure variations induce very small changes in [O₂⁻]_cyt; thus the myogenic response is unaffected by the value of β_Ca,L (which characterizes the [O₂⁻]_cyt dependence of the L-type channel conductance). In SHR, by contrast, pressure variations elicit large changes in [O₂⁻]_cyt. Hence, changes in β_Ca,L significantly modulate the arteriolar diameter in SHR, more so at 100 mmHg than at 160 mmHg (Fig. 5). Indeed, as described above, the fraction of MLC that is phosphorylated is more sensitive to changes in the cytosolic concentration of Ca²⁺ at low luminal pressures (when [Ca²⁺]_cyt is on the order of 300 nM in SHR) than at high luminal pressures (when [Ca²⁺]_cyt is >1 μM in SHR). Thus raising β_Ca,L (and therefore [Ca²⁺]_cyt) has a greater constricting effect at 100 mmHg than at 160 mmHg. As a consequence, the relative decrease in diameter following an up-step in pressure diminishes as β_Ca,L increases.

Overall, our results indicate that the parameter α_PRES has a stronger impact on model predictions than β_Ca,L. In the remaining simulations, β_Ca,L is fixed at 0.10.

Several investigators have examined the myogenic response of the afferent arteriole in SD and WKY rats. In isolated perfused hydronephrotic SD kidneys, Loutzenhiser et al. (38) measured a 36% reduction in the arteriolar diameter following a step change in pressure from 80 to 160 mmHg. Using the blood-perfused juxtamedullary nephron technique, Sharma et al. (50) found that the arteriolar diameter decreased by 11 and 27% when the pressure was increased from 100 to 130 and 160 mmHg, respectively. In other studies, however, the myogenic response was much smaller. In isolated afferent arterioles, Ren et al. (46) found that raising the luminal pressure from 80 to 140 mmHg reduced the vessel diameter by 10% in WKY; similar results were obtained in SD. Using the juxtamedullary nephron technique, Imig et al. (28) reported an 8% decrease in arteriolar diameter when the perfusion pressure was raised from 80 to 160 mmHg. With base-case parameter values, our model yields a significant myogenic response, in agreement with the first set of studies. A smaller response can be predicted if α_PRES is lowered (Fig. 4).

We also simulated the effects of adding Tempol (100 μM) on the myogenic response. In agreement with experimental observations (50), our model predicts that Tempol (100 μM) has negligible effects (<1 μm) on the myogenic response in normotensive rats, independently of the value of α_PRES.

In the study of Ren et al. (46), SHR exhibited a stronger myogenic response than WKY. The arteriolar diameter in SHR was reduced by 17–21 and 24–32% as luminal pressure was increased from 100 to 120 and 140 mmHg, respectively; these changes were accompanied by an increase in ROS production (46). Ito et al. (29) reported a comparable 34% diameter decrease in SHR when pressure was elevated from 80 to 140 mmHg. With base-case parameter values, our model predicts slightly lower values: raising luminal pressure from 100 to 120 and 140 mmHg elicits a 16 and 23% decrease in diameter, respectively (Fig. 4). If α_PRES is increased twofold, corresponding changes are 22 and 27%, i.e., within the range of the first study.

The predicted effects of Tempol on the myogenic response of SHR are shown in Fig. 6. In these simulations, α_PRES is set to either 1.75 (its base-case value) or 3.5. The model correctly predicts that Tempol significantly attenuates pressure-induced vasoconstriction in SHR. However, in contradiction with the findings of Ren et al. (46), the effects of Tempol are found to decrease with increasing luminal pressure. Possible reasons for this discrepancy are discussed below.

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DISCUSSION

We developed the present model to understand how NO and superoxide modulate the diameter of the renal afferent arteriole and its myogenic response. To do so, we expanded our previous model of the afferent arteriole vascular smooth muscle cell to include 1) the kinetics of NO and O₂⁻ formation, diffusion, and reaction, 2) the effects of NO and its second messenger cGMP on Ca²⁺ uptake and efflux into the cell, on K_Ca channel currents, and on MCLP activity, and 3) the impact of O₂⁻ on Na⁺-K⁺-ATPase pumps, SERCA, L-type Ca²⁺ channels, as well as RhoK and the NO sensitivity of sGC (Fig. 1).

We formulated a set of assumptions to obtain good agreement between model predictions and experimental measurements under a variety of conditions and for normotensive and spontaneously hypertensive rats. They key assumptions, which are based on experimental findings, are that 1) a rise in perfusion pressure elicits an increase in O₂⁻ production, and 2) O₂⁻ in turn regulates the conductance of L-type Ca²⁺ channels. In the absence of specific data, we assumed a linear dependence for both functions (Eqs. 9 and 33). With these assumptions, our model is able to adequately replicate a number of experimental findings regarding the effects of NOS inhibition, Tempol, ANG II, and variations in luminal pressure on the arteriolar diameter in normotensive and hypertensive rats, with a few caveats as discussed below.

When we simulated the full inhibition of NOS in normotensive rats, spontaneous vasomotion disappeared, whereas oscillations are maintained (55) or augmented (58) in perfused kidney preparations. One possible explanation for this discrepancy regards the effects of NO on SERCA. As illustrated in Fig. 2, when we eliminated the downstream effects of NO and cGMP one by one, oscillations were preserved, except when the effects of cGMP on the affinity of SERCA to Ca²⁺ (K_mf) were removed. We also observed that when we accounted for all the effects of NOS inhibition but set K_mf to 250 nM (vs. 307 nM in the base case, and 589 nM in the absence of cGMP), spontaneous vasomotion was preserved; the arteriolar diameter oscillated around 15.8 μm with an amplitude of 1.3 μm. It is therefore possible that the model does not properly account for the effects of cGMP on SERCA, i.e., that findings in rat aortic smooth muscle cells (5) cannot be extrapolated to renal afferent arterioles.

Validating the model by comparing its myogenic response predictions with in vivo or in vitro data is made difficult by the variability of experimental results. As described in model results, the magnitude of the myogenic response in SD and WKY rats differs significantly between studies, even when similar preparations are used (28, 38, 46, 50). Moreover, whether SHR have a stronger response than normotensive rats remains unclear. Studies in isolated afferent arterioles suggest that pressure-induced constriction is enhanced in SHR (29, 46), but other investigations, performed on isolated perfused kidneys, indicate that the autoregulatory response of the afferent arteriole is similar in SHR and WKY (21, 45). Additionally, several investigations indicate that Tempol induces vasodilation of the afferent arteriole in SHR at normal perfusion pressures (8, 17, 26), but at least one study found no effect of Tempol on the basal arteriolar diameter of these rats (46).

As we previously discussed (11), the mechanisms underlying mechanotransduction in vascular smooth muscle cells have yet to be fully understood. The nature of the channels that sense variations in luminal pressure remains uncertain. Our model assumes that pressure changes first activate a channel that carries predominantly Na⁺ and Ca²⁺, with a 20:3 Na⁺-to-Ca²⁺ conductance ratio. The pressure-dependence of this conductance is characterized by a proportionality constant α_Pres. Our results indicate that, for a given value of α_Pres, fractional changes in arteriolar diameter induced by pressure elevations are smaller in SHR than in normotensive rats. This is because the sensitivity to [Ca²⁺]_cyt variations of the phosphorylated fraction of MLC decreases with increasing [Ca²⁺]_cyt and basal values of [Ca²⁺]_cyt are significantly higher in SHR than in normotensive rats. If indeed the myogenic response of SHR is stronger than that of WKY, our simulations suggest that this could be explained by a higher conductance of mechanosensitive channels in the former.

Our model predicts that Tempol has a mild impact on the myogenic response of SHR at high luminal pressures, whereas Ren et al. (46) observed that Tempol exerted greater effects at large pressure values. In our model, the mechanosensitive channel conductance is unaffected by the presence of Tempol and remains elevated at high pressures; as a result, intracellular Ca²⁺ levels stay high, which favors vasoconstriction. Thus this discrepancy could indicate that mechanosensitive channels are sensitive to ROS and/or NO levels. There is evidence that O₂⁻ and NO, most likely via their product peroxynitrite, modulate mechanosensitive channels in ventricular myocytes (9), but the underlying mechanisms have not been determined. If we assume that the mechanosensitive channel conductance increases when [O₂⁻]_cyt is elevated, the myogenic response in SHR is then enhanced, but this does not affect the predicted arteriolar diameter in the presence of Tempol. Conversely, if we assume that the mechanosensitive channel conductance is significantly reduced when [O₂⁻]_cyt is low (<1 nM), adding Tempol leads to greater vasodilation in SHR; however, if the mechanosensitive channel conductance is similarly reduced in normotensive rats, Tempol should then exert significant vasodilatory effects in these rats too, which contradicts experimental findings. Further experimental characterization of mechanosensitive channels and their regulation by NO and ROS would be helpful. We should also acknowledge that our assumption that SHR and normotensive rats differ only by their superoxide levels may be too simplistic; other differences between the two strains, such as varying levels of cyclooxygenase-derived products (16), are likely to influence their contractile responses.

The cytosolic concentration of superoxide in vascular smooth muscle cells has not been measured, to our knowledge. We adjusted its production rate to obtain O₂⁻ concentrations in the nanomolar range (Table 3) and took into consideration the effects of ANG II and perfusion pressure on O₂⁻ synthesis, but we should recognize this uncertainty. Our estimate of the plasma membrane permeability to NO (P_NO) is equally unsure. Based on the measured value of the NO diffusion coefficient in octanol (40), and assuming a membrane thickness of 1 μm, P_NO is estimated as 0.03 cm/s. With this hypothesis, the model predicts that, in normotensive rats, most of the NO that enters the afferent arteriole smooth muscle cell from the endothelium diffuses out on the other side. The fraction of NO that instead reacts with O₂⁻ is 28% in normotensive rats, vs. 53% in SHR (at 100 mmHg). Conversely, the fraction of O₂⁻ that is scavenged by NO and SOD is, respectively, 35 and 55% in normotensive rats, and 26 and 62% in SHR. In comparison, in the rabbit aorta, the fraction of produced NO that is consumed by chemical reaction has been estimated as 37% (40).

**Table 3. Key model predictions for normotensive and spontaneously hypertensive rats**
	SD			SHR
	Base case	140 mmHg	NOS inhibition	Base case	140 mmHg	NOS inhibition
AA diameter, μm	20.5 (1.9)	14.3	16.0	17.1	13.1	14.5
[NO]_cyt, nM	152	147	0	100	76	0
[O₂⁻]_cyt, nM	4.9	5.6	7.6	14.0	22.1	18.9
[cGMP], μM	11.1	11.0	0.027	5.3	4.5	0.027
$G_{O_{2}^{-}}$ , μM.s⁻¹	14.4	16.1	14.4	36.0	53.3	36.0
k₂^Myo, s⁻¹	0.150	0.108	0.118	0.113	0.085	0.096
G_Ca,L, nS	2.75	2.78	2.89	3.24	3.69	3.51
G_Ca,Pres⁰, nS	0.0200	0.0527	0.0200	0.0200	0.0527	0.0200
V_KCa, mV	88.7 (10.8)	75.5	192.9	97.0	84.5	189.7
I_PMCA,max, pA	4.96	4.96	4.55	4.92	4.91	4.55
K_mf, nM	307	307	574	320	324	574

Numbers in parenthesis indicate amplitude when oscillations are present.

AA, afferent arteriole; NOS, nitric oxide synthase.

See mathematical model for the meaning of other parameters.

Based on experimental evidence, the model assumes that O₂⁻ modulates the current across Na⁺-K⁺-ATPase and SERCA only when its concentration is in the micromolar range; in the scenarios examined here, [O₂⁻]_cyt remains below 100 nM and therefore has negligible effects on these two currents. Under these conditions, O₂⁻ acts in the following ways: by modulating the conductance of L-type Ca²⁺ channels, by activating RhoK (which in turn inactivates MLCP), by scavenging NO, and by lowering the responsiveness of sGC to NO; the latter two mechanisms counteract NO-mediated changes in [Ca²⁺]_cyt and MLC phosphorylation. As shown in Table 3, the NO-independent effects of O₂⁻ seem to predominate in the scenarios we considered: per se (e.g., at a given perfusion pressure), changes in O₂⁻ production are predicted to have a much greater impact on G_Ca,L than on NO signaling pathways. Stated differently, if the rate of NO scavenging by O₂⁻ were the same in SHR as in normotensive rats, the SHR afferent arteriolar diameter would only be 0.5 μm higher than its base-case value; i.e., the reduced diameter in SHR does not appear to stem mostly from NO scavenging. These predictions accord with experimental observations suggesting that elevated renal resistance in SHR is unlikely to stem from ROS-induced quenching of NO (8).

The present model represents a single afferent arteriole smooth muscle cell with a baseline luminal pressure of 100 mmHg. However, luminal hydrostatic pressure decreases along the afferent arteriole, from ∼100 mmHg at the inlet to ∼50 mmHg at the glomerular end. To represent a segment of the afferent arteriole that is closer to the glomerulus and is thus typically affected by a substantially lower hydrostatic pressure, the myogenic response (Eq. 18) could be modified. More specifically, the parameters P₁ and P₂ in Eq. 18 could be chosen as a function of baseline pressure, so that the same absolute pressure change would induce the same change in conductance (G_i,Pres), regardless of the baseline hydrostatic pressure. This approach allows one to represent smooth muscle cells at differing locations of the afferent arteriole. A model of an afferent arteriole could then be built by connecting a series of these cell models, coupled via their gap junctions, as in Ref. 49. In conclusion, our mathematical model supports the hypothesis that, per se, increased O₂⁻ production in the afferent arteriole of SHR substantially alters the myogenic response of the arteriole. Our results also suggest that the constrictive effects of superoxide are not predominantly due to its scavenging of NO, but mostly stem from the NO-independent action of O₂⁻ on L-type Ca²⁺ channels and RhoK signaling.

GRANTS

This research was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-089066 and National Science Foundation Grant DMS-1263995.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: A.T.L. and A.E. conception and design of research; A.T.L. and A.E. performed experiments; A.T.L. and A.E. analyzed data; A.T.L. and A.E. interpreted results of experiments; A.T.L. and A.E. prepared figures; A.T.L. and A.E. drafted manuscript; A.T.L. and A.E. edited and revised manuscript; A.T.L. and A.E. approved final version of manuscript.

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AUTHOR NOTES

Address for reprint requests and other correspondence: A. Edwards, Centre de Recherche des Cordeliers, 15 rue de l'Ecole de Médecine, 75006 Paris, France (e-mail: aurelie.[email protected]jussieu.fr).

American Journal of Physiology-Renal Physiology 309 8 cover image

Volume 309Issue 8October 2015Pages F708-F719

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History

Received 1 May 2015

Accepted 9 July 2015

Published online 15 October 2015

Published in print 15 October 2015

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Predicted effects of nitric oxide and superoxide on the vasoactivity of the afferent arteriole

Abstract

MATHEMATICAL MODEL

Ion and Charge Conservation Equations

Kinetics of Formation of NO and O2−

Nitric oxide.

Superoxide.

Soluble Guanylate Cyclase Activation and cGMP Generation

Activation of soluble guanylate cyclase by NO.

Generation and degradation of cGMP.

Mechanosensitive Channels

NO and cGMP Effects on K+ Transport Pathways

O2− Effects on K+ and Na+ Transport Pathways

Effects of cGMP, NO, and O2− on Ca2+ Transport Pathways

MLCK- and MLCP-Dependent Phosphorylation of Myosin

Vessel Mechanics

MODEL RESULTS

Effects of Superoxide on the Basal Diameter of the Afferent Arteriole

Effects of NO Synthase Inhibition and Tempol

Effects of ANG II

Effects of Luminal Pressure Increases

DISCUSSION

GRANTS

DISCLOSURES

AUTHOR CONTRIBUTIONS

REFERENCES

AUTHOR NOTES

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Kinetics of Formation of NO and O₂⁻

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Effects of cGMP, NO, and O₂⁻ on Ca²⁺ Transport Pathways