Impact of nitric-oxide-mediated vasodilation and oxidative stress on renal medullary oxygenation: a modeling study

Abstract

The goal of this study was to investigate the effects of nitric oxide (NO)-mediated vasodilation in preventing medullary hypoxia, as well as the likely pathways by which superoxide (O₂⁻) conversely enhances medullary hypoxia. To do so, we expanded a previously developed mathematical model of solute transport in the renal medulla that accounts for the reciprocal interactions among oxygen (O₂), NO, and O₂⁻ to include the vasoactive effects of NO on medullary descending vasa recta. The model represents the radial organization of the vessels and tubules, centered around vascular bundles in the outer medulla and collecting ducts in the inner medulla. Model simulations suggest that NO helps to prevent medullary hypoxia both by inducing vasodilation of the descending vasa recta (thus increasing O₂ supply) and by reducing the active sodium transport rate (thus reducing O₂ consumption). That is, the vasodilative properties of NO significantly contribute to maintaining sufficient medullary oxygenation. The model further predicts that a reduction in tubular transport efficiency (i.e., the ratio of active sodium transport per O₂ consumption) is the main factor by which increased O₂⁻ levels lead to hypoxia, whereas hyperfiltration is not a likely pathway to medullary hypoxia due to oxidative stress. Finally, our results suggest that further increasing the radial separation between vessels and tubules would reduce the diffusion of NO towards descending vasa recta in the inner medulla, thereby diminishing its vasoactive effects therein and reducing O₂ delivery to the papillary tip.

despite receiving ∼25% of the cardiac output, the mammalian kidney (particularly the medulla) is susceptible to hypoxia. Blood flow in the renal cortex is high, and its filtration by glomerular capillaries creates a need for tubular transport processes so as to maintain homeostasis. Partly because of the need to concentrate urine, medullary blood flow is substantially lower than cortical blood flow. The resulting low oxygen (O₂) delivery to the outer medulla (OM) yields an oxygen tension (Po₂) of 20 mmHg, compared with 50 mmHg in the cortex (7, 38). Medullary hypoxia is believed to be an important pathway in the development of chronic kidney diseases and, in general, is due to a mismatch between changes in renal O₂ supply and consumption (16).

Nitric oxide (NO) and superoxide (O₂⁻) exert opposite effects on O₂ supply and consumption in the renal medulla, such that medullary oxygenation is partly governed by the balance between the two solutes. NO induces vasodilation of descending vasa recta (DVR) (43), thereby enhancing blood flow into, and thus O₂ supply to, the medulla; on the other hand, O₂⁻ favors reductions in medullary blood flow (12). O₂ consumption in the renal medulla mostly stems from the active transport of sodium across tubular epithelia; whereas Na⁺ reabsorption in the medullary thick ascending limb (mTAL) is stimulated by O₂⁻ (40), it is inhibited by NO (41).

We previously examined the impact of NO and O₂⁻ on salt transport and O₂ consumption (17) but without accounting for their effects on blood flow and O₂ supply. The main objective of the present study was to investigate the degree to which modulation of DVR muscle tone by NO helps to protect the kidney from medullary hypoxia, via increases in blood flow into the medulla and thus O₂ delivery.

NO induces vasodilation of medullary DVR by acting on the pericytes that surround the vascular endothelium. These vascular smooth muscle cells impart contractile properties to the vessels, such that increases in NO reduce intracellular Ca²⁺ via the cGMP/PKG pathway and cause vasodilation (14). Kakoki et al. (25) found that when NOS inhibitors were administered near the OM-inner medullary (IM) boundary at doses lowering NO levels by 32 and 66%, respectively, medullary blood flow was reduced by 32 and 40% as a result. Mattson et al. (33) observed that infusion of NOS stimulators into the renal medullary interstitium selectively increased medullary blood flow, without affecting superficial cortical blood flow.

Anatomical studies of rat kidneys have shown that the radial organization of tubules and vessels in the medulla is highly structured (32, 45, 46, 47, 48). In particular, in the inner stripe, O₂-supplying DVR are isolated within tightly packed vascular bundles, distant from the mTALs and collecting ducts (CDs); in the upper IM, CDs form clusters separate from DVR. Recent modeling studies have demonstrated that this architecture plays a role in medullary oxygen distribution and hypoxia (9, 10, 18, 19). Another goal of the present study was to examine anew the impact of the structural organization of nephrons and vessels in the renal medulla on the distribution of O₂, accounting for modulation of Po₂ by NO and superoxide. To accomplish this, we expanded a detailed mathematical model of solute transport in the rat renal medulla to include effects of NO on DVR diameter and flow.

MODEL DESCRIPTION

We extended a detailed model of solute transport and oxygenation in the renal medulla of the rat kidney (17) to include the effects of NO-induced vasodilation. The model accounts for the three-dimensional architecture of the renal medulla of a rat kidney (27, 29, 47). Particularly relevant to oxygen transport is the sequestration of the vasa recta within the vascular bundles in the inner stripe of the outer medulla and the separation of the thick ascending limbs from the oxygen-rich vascular bundles (3, 27), both of which are captured in the model (see Fig. 1, reproduced from Ref. 17).

Fig. 1.

Schematic diagram of a cross section through the outer stripe, inner stripe, upper inner medullary (IM), mid-IM, and deep IM, showing interstitial regions (R1, R2, R3, R4) and relative positions of tubules and vessels. Decimal numbers indicate relative interaction weightings with regions [e.g., in the outer stripe, half of the short descending vasa recta (SDV) lie in R1, and half lie in R2]. SDL/SAL, descending/ascending limbs of short loops of Henle; LDL/LAL, descending/ascending limbs of long loops of Henle; subscript “S,” “M,” and “L” associated with a LAL denotes limbs that turn with the 1st millimeter of the IM (S), within the mid-IM (M), or reach into the deep IM (L); CD, collecting duct; SDV, short descending vasa recta (DVR); LDV, long DVR; AVRs, populations of ascending vasa recta. Dotted-line box in deep IM indicates that LDV, LDL, LAL, and CD are weighted evenly between the 4 regions. Tubules, vessels, and interstitium are denoted in blue, red, and pink, respectively. [Reproduced from Ref. (17).]

The model represents DVR that supply blood to every depth of the renal medulla and loops of Henle that turn at every depth. At all depths along the cortico-medullary axis, a fraction of the DVR is assumed to peel off and supply the capillary plexus. The number ratio between DVR that supply the OM vs. the IM (denoted SDV and LDV below) is assumed to be 44:12 (30). To represent the relatively more abundant capillary flow in the inner stripe, the population distribution of SDV is given by

$$w_{\text{SDV}}\left(x\right)=\{\begin{array}{ll}1-0.1x/L_{\mathrm{\text{OS}}},\hfill & 0\le x\le L_{\text{OS}},\hfill \\ \left(1-0.1L_{\text{OS}}/L_{\text{OM}}\right)\left(1-\frac{x-L_{\text{OS}}}{L_{\mathrm{O}\mathrm{M}}-L_{\mathrm{O}\mathrm{S}}}\right),\hfill & L_{\text{OS}}\le x\le L_{\text{OM}},\hfill \end{array}$$ (1)

where L_OS and L_OM denote the length of the outer stripe and OM, respectively. The population of the DVR that reach into the IM is given by

$$w_{\text{LDV}}\left(x\right)=\{\begin{array}{ll}1,\hfill & 0\le x\le L_{\text{OM}},\hfill \\ \left(1-0.95\left(\frac{x-L_{\text{OM}}}{L_{\text{IM}}}\right)\right)e^{{-3.5}^{((\mathit{x}-L_{\text{OM})}/L_{\text{IM}})}},\hfill & L_{\text{OM}}\le x\le L_{\text{IM}},\hfill \end{array}$$ (2)

where L_IM denotes the length of the IM.

The model solution describes flows and solute concentrations as a function of medullary depth in the interstitium, the loops of Henle, CDs, vasa recta, and capillaries. Blood flow in vasa recta is divided into two compartments, plasma and red blood cells (RBCs), which interact with each other; since plasma flow and RBC flow are computed separately, the hematocrit in vasa recta, given by the ratio of RBC flow to total blood flow, varies with medullary depth. The solutes explicitly considered in the model are NaCl, urea, deoxy- and oxyhemoglobin (Hb and HbO₂), O₂, NO, HbNO, and O₂⁻. Detailed transport equations can be found in our previous study (17) and the references therein. We focus here on the new feature of our model, namely, NO-induced vasodilation of DVR. We also briefly summarize the equations describing the kinetics of formation of NO and O₂⁻ and active Na⁺ transport and O₂ consumption in the medulla.

NO-induced vasodilation.

Wrapping around the DVR endothelium are sporadically occurring pericytes, which are vascular smooth muscle cells and thus can induce vasoconstriction or dilation. NO is believed to induce vasodilation by reducing cytosolic Ca²⁺ concentration ([Ca²⁺]) and the fraction of bound cross-bridges, via the cGMP/PKG pathway (14). Here we adopt a phenomenological approach and assume that the DVR radius (R_DVR, in μm) increases with local increases in NO concentration (C̄_NO^DVR). The model represents DVR of differing length, and at medullary depth x, the radius of a DVR (which we assume cannot fall below that of a single erythrocyte, ∼3.5 μm) is given by

$$\frac{R_{\text{DVR}}\left(x\right)}{R^{\ast }}=\max \left\{3.5,\frac{A\left[{\overline{C}}_{\text{NO}}^{\text{DVR}}\left(x\right)/C_{\text{NO}}^{\ast }\right]}{\left(A-1\right)+\left[{\overline{C}}_{\text{NO}}^{\text{DVR}}\left(x\right)/C_{\text{NO}}^{\ast }\right]}\right\},$$ (3)

where x ranges between 0 and the length of the medulla (7 mm); ${\overline{C}}_{\text{NO}}^{\text{DVR}}\left(x\right)$ denotes the average of the DVR plasma and local interstitial [NO] at x; $C_{NO}^{*}$ denotes the reference NO concentration, taken to be 25 nM, which is the average medullary interstitial concentration predicted in our previous model (17); R* denotes the reference radius, taken to be 7.5 μm; and A is a dimensionless parameter, chosen to be 1.0962 such that baseline DVR volume flow at the cortico-medullary boundary is ∼8 nl/min (4). With our choice of parameters, decreases in NO levels by 32 and 66% lead to corresponding decreases in R_DVR of 10 and 12% and decreases in blood flow into the medulla by 32 and 40%, matching the experimental data of Kakoki et al. (25).

We assume that medullary blood flow can be described by Poiseuille's law (1, 21, 34), so that blood flow along each DVR is determined, in part, by the pressure drop (ΔP_DVR) and vascular resistance; the latter scales inversely with the fourth power of the DVR radius:

$$F_{\text{V}}^{\text{DVR}}=\frac{\pi }{8}\frac{\Delta P_{\text{DVR}}\cdot R_{\text{DVR}}^4}{\mu \cdot L_{\text{DVR}}},$$ (4)

where F_V^DVR is blood flow in an individual DVR, μ is the blood viscosity, and L_DVR denotes vessel length. As previously noted, the DVR terminate at every medullary depth; thus L_DVR varies from 0 to 7 mm, the length of the medulla. Because the DVR have differing lengths and radii that vary in space, blood flow distribution may be inhomogeneous among the DVR. To determine entering blood flow for the DVR, we connect each DVR to a downstream resistance in series (which varies among DVR of differing lengths), and we assume that pressures at the two ends (efferent arteriole and renal vein, denoted P_E and P_V, respectively) are known a priori. We then assume that the efferent arteriole-to-renal vein pressure drop (P_E − P_V) is equal to the pressure drop along that DVR and along its downstream resistance (Γ_down):

$$P_{\text{E}}-P_{\text{V}}=\Delta P_{\text{DVR}}+F_{\text{V}}^{\text{DVR}}\left(L_{\text{DVR}}\right){\Gamma }_{\text{down}}.$$ (5)

F_V^DVR(L_DVR) denotes volume flow at the end of the DVR, and Γ_down is a function of DVR length. A schematic of the downstream resistance and the distributed formulation of the DVR is shown in Fig. 2. The pressure drop ΔP_DVR is computed along the length of the DVR:

$$\Delta P_{\text{DVR}}=\frac{8\mu }{\pi }\left[{\displaystyle {\int }_0^{L_{\text{DVR}}}}\frac{F_{\text{V}}^{\text{DVR}}\left(s\right)}{R_{\text{DVR}}^{\text{4}}\left(s\right)}ds\right].$$ (6)

where μ is taken to be 2 cP. The downstream resistance, Γ_down, is chosen so that under baseline conditions, DVR inlet blood flow is 8 nl/min. Γ_down is then assumed fixed under all other conditions.

Fig. 2.

Schematic diagram illustrating the distributed formulation of the DVR and the downstream resistance, from Eq. 5. A fraction of the DVR turn at every depth of the corticomedullary axis, and red blood cells within the DVR (DRBC) are represented by rigid tubes and interact with surrounding plasma. Each DVR is assumed connected upstream to an efferent arteriole (with pressure P_E) and to a downstream resistance that connects to the renal vein (with pressure P_V).

Generation of NO and O₂⁻.

NO is produced primarily by endothelial cells, although epithelial cells also contribute to its synthesis. Once generated, NO diffuses into the surrounding interstitium or the lumen, where it is partly consumed by several scavengers, including O₂⁻, HbO₂, and Hb. The production rate of NO, denoted G_i,NO, is assumed to depend on local Po₂, such that for tubule or vessel i

$$G_{i,\text{NO}}=G_{i,\text{NO}}^{\max }\left[\frac{C_{i,{\text{O}}_2}\left(x\right)}{K_{{\text{O}}_2}^{\text{NO}}+C_{i,{\text{O}}_2}\left(x\right)}\right],$$ (7)

where G_i,NO^max is the maximal rate of NO production (fixed in a particular tubule or vessel i, see Table 2 in Ref. 17); C_i,O2 is the luminal O₂ concentration in tubule or vessel i, and K_O2^NO is the Michaelis constant (set to 50.9 μM in all structures; Ref. 54).

Similarly, the dependence of O₂⁻ generation rate on Po₂ is given by

$$G_{i,{\text{O}}_{\text{2}}^{-}}=G_{i,{\text{O}}_{\text{2}}^{-}}^{\max }\left[\frac{C_{i,{\text{O}}_2}\left(x\right)}{K_{{\text{O}}_{\text{2}}}^{{\text{O}}_{\text{2}}^{-}}+C_{i,{\text{O}}_2}\left(x\right)}\right],$$ (8)

where $G_{i,{\text{O}}_{\text{2}}^{-}}^{\max }$ is the maximal rate of O₂⁻ production (fixed in a particular tubule or vessel i, see Table 2 in Ref. 17) and $K_{{\text{O}}_{\text{2}}}^{{\text{O}}_{\text{2}}^{-}}$ is the Michaelis constant (set to 20.6 μM in all structures; Ref. 11). Once generated, O₂⁻ then exits the cell via diffusion or reacts with scavengers (NO or superoxide dismutase).

Although oxygen is a precursor to NO synthesis, experiments have demonstrated that medullary NO levels increase in the presence of hypoxia (22, 35, 42, 49). We account for hypoxia-induced stimulation of NO release in a simplified manner: by assuming that as the availability of O₂ decreases, the permeability of the RBCs to NO (P_NO^RBC) also decreases, thereby preserving more NO in the medullary tissue. Specifically, we assume that P_NO^RBCvaries linearly with Po₂

$$P_{\text{NO}}^{\text{RBC}}=P_{\text{NO}}^{\text{RBC,basal}}\left[\frac{1}{2}+\frac{1}{2}\left(\frac{P{\mathrm{o}}_2-P_{\text{a}}}{P_{\text{b}}-P_{\text{a}}}\right)\right]$$ (9)

where P_NO^RBC,basal is the basal RBC permeability to NO, taken here to be 0.1 cm/s (15). The parameters in Eq. 9 were chosen in a previous study (15) to yield an increase in medullary NO levels consistent with experimental observations in anesthetized rats administered indomethacin (22a) when Po₂ levels decreased from P_b = 28 mmHg to P_a = 12 mmHg.

Active Na⁺ transport and O₂ consumption.

Active transport of Na⁺ occurs along some tubular segments, including the thick ascending limbs, proximal straight tubules, and CDs. The rate of active Na⁺ transport is assumed to depend on local Po₂, [NO], and [O₂⁻], and is expressed assuming Michaelis-Menten kinetics as follows:

$${\Psi }_{i,\text{Na}}^{\text{active}}=V_{\max ,i,\text{Na}}\left[\frac{C_{i,\text{Na}}\left(x\right)}{K_{\text{M,Na}}+C_{i,\text{Na}}\left(x\right)}\right]\cdot f_{\text{act}}\left(C_{i,{\text{O}}_2}\right)\cdot g_{\text{act}}\left(C_{i,\text{NO}}\right)\cdot h_{\text{act}}\left(C_{i,{\mathrm{O}}_2^{-}}\right),$$ (10)

where V_max,i,Na (in nmol Na⁺·cm⁻²·s⁻¹) is the maximal rate of Na⁺ transport (see Table 1 in Ref. 29); K_M,Na is the Michaelis constant (taken here to be 70 mM; Refs. 20, 31); C_i,Na(x) is the luminal Na⁺ concentration in tubule i, and the functions f_act(C_i,O2), g_act(C_i,NO), and h_act$\left(C_{i,{\text{O}}_{\text{2}}^{-}}\right)$ represent the effects of O₂, NO, and O₂⁻ on the rate of active sodium reabsorption.

Under well-perfused conditions, active Na⁺ transport is not limited by O₂ levels (37). Below some Po₂ threshold (P_i,c, set to 10 mmHg in all tubular segments; Ref. 18), aerobic transport decreases and is partially compensated by anaerobic metabolism. To describe this process, we set

$$f_{\text{act}}\left(C_{i,{\text{O}}_2}\right)=\{\begin{array}{ll}1\hfill & {\text{if Po}}_{\text{2}}\ge P_{i,c}\hfill \\ \frac{{\text{Po}}_2}{P_{i,c}}\left(1-F_{\text{AN}}\right)\hspace{0.17em}+\hspace{0.17em}{F}_{\mathrm{A}\mathrm{N}}\hfill & {\text{if Po}}_{\text{2}}<P_{i,c,}\hfill \end{array}$$ (11)

where F_AN describes the capacity of the tubular segment for anaerobic transport and is taken to be 0.5 in the OMCD (52, 55), 0.4 in the IMCD (51), 0.1 in the TALs (2), and 0.14 in the proximal straight tubules (13).

NO and O₂⁻ have opposite effects on Na⁺ transport, with NO having an inhibitory effect and O₂⁻ enhancing Na⁺ transport. These effects are represented, respectively, by

$$g_{\text{act}}\left(C_{i,\text{NO}}\right)=1-\frac{C_{i,\text{NO}}\left(x\right)}{{\beta }_i+C_{i,\text{NO}}\left(x\right)},$$ (12)

$$h_{\text{act}}\left(C_{i,{\text{O}}_{\text{2}}^{-}}\right)=0.7+0.6\left(\frac{C_{i,{\text{O}}_{\text{2}}^{-}}\left(x\right)}{{\gamma }_i+C_{i,{\text{O}}_{\text{2}}^{-}}\left(x\right)}\right),$$ (13)

where C_i,NO(x) and $C_{i,{\text{O}}_{\text{2}}^{-}}$(x) are the local NO and O₂⁻ concentrations in tubule or vessel i and β_i and γ_i are the Michaelis constants (set to 47 nM and 0.2 pM in the TALs and 232 nM and 0.06 pM in the CDs, respectively; Ref. 17).

To determine the oxygen consumption arising from active Na⁺ transport, we associate with tubule i transport a TQ_i value, which is the number of moles of Na⁺ actively reabsorbed per mole of O₂ consumed under maximum efficiency. Then the volumetric rate of active O₂ consumption in tubule i is given by

$$R_{i,{\text{O}}_2}^{\text{active}}=\frac{2\pi r_i}{A_i^{\text{epi}}}\frac{{\psi }_{i,\text{Na}}^{\text{active}}}{{\text{TQ}}_i}\Theta \left(P_{i,{\text{O}}_2}\right),$$ (14)

where r_i is the inner radius of tubule i, A_i^epi is the cross-sectional area of the surrounding epithelium of tubule i and Θ(P_i,O2) is the fraction of remove Na⁺ active transport supported by aerobic respiration. We assume that below a certain threshold, aerobic respiration decreases linearly, so that Θ(P_i,O2) is given by

$$\Theta \left(P_{i,{\text{O}}_2}\right)=\{\begin{array}{ll}1\hfill & {\text{if Po}}_{\text{2}}\ge P_{i,c}\hfill \\ \frac{{\text{Po}}_2/P_{i,c}}{F_{\text{AN}}+\left(1-F_{\text{AN}}\right)\left({\text{Po}}_2/P_{i,c}\right)}\hfill & {\text{if Po}}_{\text{2}}<P_{i,c^{\mathrm{.}}}\hfill \end{array}$$ (15)

In addition, a basal level of O₂ consumption from processes other than Na⁺ active transport is assumed to take place in the vascular endothelial cells, tubular epithelial cells, and interstitial cells. This rate of basal O₂ consumption in tubule or vessel i is given by

$$R_{i,{\text{O}}_2}^{\text{basal}}=R_{\max ,{\text{O}}_2}^{\text{basal}}\left[\frac{C_{i,{\text{O}}_2}\left(x\right)}{K_{\text{M},i}^{{\mathrm{O}}_2}+C_{i,{\text{O}}_2}\left(x\right)}\right],$$ (16)

where K_M,i^O₂ is the Michaelis constant and R_max,O2^basal is the maximal volumetric rate of O₂ consumption (17).

The model assumes that the DVR deliver oxygen both in the form of O₂ dissolved in blood and of HbO₂ bound to hemoglobin in RBCs. The reaction between Hb and O₂ is assumed to be at equilibrium, as the kinetics of Hb O₂ dissociation are substantially faster than the diffusion of O₂ (11a). Oxyhemoglobin saturation (So₂) is assumed to be well fitted by the Hill equation, so that So₂ is given by

$$\frac{C_{{\text{HbO}}_2}^{\text{RBC}}}{C_{{\text{HbO}}_2}^{\text{RBC}}+C_{\text{Hb}}^{\text{RBC}}}=\frac{{\left(C_{{\text{O}}_2}^{\text{RBC}}\right)}^{\mathit{\text{n}}}}{{\left(C_{50}\right)}^{\text{n}}+{\left(C_{{\text{O}}_2}^{\text{RBC}}\right)}^{\mathit{\text{n}}}},$$ (17)

where n is the exponent of the Hill equation (set to 2.6), and C₅₀ is the O₂ concentration at which So₂ is 0.5, taken to be 41.2 μM (11a). As such, the oxy- and deoxyhemoglobin concentration changes with medullary depth, due to loading and unloading of O₂.

RESULTS

Base case results.

To assess the impact of NO on vascular tone, we compared two cases throughout this study: a vasoactive case (represented with white bars on figures), in which the effects of NO on DVR diameter and flow were considered; and a fixed resistance case (represented with black bars), in which DVR diameter was fixed independently of [NO].

Without the vasoactive influence of NO, DVR radius is fixed at 7.5 μm. The model predicts a medullary O₂ supply (i.e., the total amount of oxygen in the blood entering the medulla) of 27.9 pmol·min⁻¹·nephron⁻¹, a total (active plus basal) mTAL O₂ consumption rate (Qo₂) of 15.8 pmol·min⁻¹·nephron⁻¹, and an average superficial mTAL (SAL) luminal Po₂ of 7.8 mmHg. (We report SAL Po₂ because it tends to represent the lowest medullary O₂ levels, as well as total mTAL Qo₂ because it gives a more complete picture of medullary O₂ consumption.) Of the medullary O₂ supply, 58.4% is consumed in the OM, and 16.5% is delivered to the IM. When DVR vasoactivity due to NO is included, the length-averaged radius over all DVR is slightly higher, at 7.7 μm, resulting in a slightly higher medullary O₂ delivery of 30.1 pmol·min⁻¹·nephron⁻¹. Of that delivery to the medulla, 60.1% is consumed in the OM, and 16.1% is delivered to the IM. The mTAL Qo₂ also increases slightly to 16.8 pmol·min⁻¹·nephron⁻¹. These competing factors result in a net increase in SAL Po₂, to 8.3 mmHg. In the simulations that follow, we examined several scenarios involving alterations of medullary NO and superoxide effects.

Effects of NO on medullary oxygenation.

It has been well established that NO helps to blunt hypoxia and keep Po₂ levels sufficiently high (6). The NO-mediated Po₂ increase may be attributed to increased O₂ supply (from dilated blood vessels) or decreased O₂ consumption (from inhibition of active sodium transport). To assess the relative contributions of the two pathways, we first conducted a simulation in which we removed the vasodilatory effect of NO by setting R_DVR to a fixed value (5.9 μm) given by Eq. 3 when ${\overline{C}}_{\text{NO}}^{\text{DVR}}(x)$ is set to 6.3 nM, which corresponds to one-fourth of the reference value of C_NO^*. Note that [NO] was fixed in Eq. 3 but not elsewhere. That is, [NO] was determined locally based on conservation equations. The DVR radius, predicted medullary O₂ supply, mTAL Qo₂, and SAL luminal Po₂ are shown in Fig. 3. This case yielded an average DVR radius that is 20.8% smaller than in the base case, resulting in a 155% increase in vascular resistance. Consequently, medullary O₂ supply decreased by 61% (Fig. 3B), and SAL Po₂ decreased by 28% (Fig. 3D).

Fig. 3.

Average DVR radius (μm; A), medullary O₂ supply (pmol·min⁻¹·nephron⁻¹; B), medullary thick ascending limb (mTAL) O₂ consumption (pmol·min⁻¹·nephron⁻¹; C), and average SAL Po₂ (mmHg; D), in the base case, with effects of ${\overline{C}}_{\text{NO}}^{\text{DVR}}(x)$ = ¼C_NO^* on DVR radius and with effects of C_i,NO = ¼C_NO^* on g_act.

In a second simulation, we lowered instead the (inhibitory) effect of NO on the rate of active Na⁺ transport $\left({{\mathrm{\Psi }}_{\mathrm{N}\mathrm{a}}^{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}}_{}^{}\right)$ by setting g_act to a fixed value given by Eq. 12 when C_i,NO is set to ¼C_NO^*. Key model predictions are shown in Fig. 3. When thus reducing the inhibitory effects of NO on sodium reabsorption, mTAL Qo₂ increased by 62%. As a result, average SAL luminal Po₂ dropped substantially, from the baseline value of 8.3 mmHg to 4.9 mmHg. A comparison between these two cases (Fig. 3D) suggests that the inhibitory effect of NO on ${\mathrm{\Psi }}_{Na}^{active}$, the reduction of which resulted in a 41% decrease in SAL Po₂, has a greater influence on mTAL Po₂ than does the vasodilatory effect of NO on DVR radius, the reduction of which resulted in a 28% decrease in SAL Po₂.

We then examined the effects of varying NO generation rate: the latter was either reduced by 75% or doubled. For each case, we conducted simulations with and without [NO]-induced vasodilation. For the latter, DVR radius was set to the reference radius value of 7.5 μm (independent of [NO]).

Corresponding results are summarized in Fig. 4. The model predicts that reducing the rate of NO synthesis lowers medullary O₂ supply and raises O₂ consumption by comparable amounts. The fact that low NO levels lead to significant vasoconstriction means that the vasoactive properties of the DVR exacerbate the low Po₂ conditions when NO is reduced. Conversely, increasing the rate of NO synthesis raises medullary O₂ supply and lowers O₂ consumption to similar extents. In other words, sufficiently high NO concentrations lead to higher SAL Po₂ levels when allowing for NO-induced vasodilation. Altogether these results indicate that the vasoactive effects of NO contribute significantly to medullary oxygenation.

Fig. 4.

Average DVR radius (μm; A), medullary O₂ supply (pmol·min⁻¹·nephron⁻¹; B), mTAL O₂ consumption (pmol·min⁻¹·nephron⁻¹; C), and average SAL Po₂ (mmHg; D), in the base case, with G_NO^max × 25%, and with G_NO^max × 200%, both with (white bars) and without (black bars) [NO]-induced vasodilation.

Oxidative stress and medullary oxygenation.

Oxidative stress in the kidney is associated with a reduction in tissue oxygenation and can be caused by an increase in superoxide levels. To simulate oxidative stress in this study, we increased the superoxide generation rate, $G_{i,{\text{O}}_{\text{2}}^{-}}^{\max }$, by a factor of 2. Higher superoxide levels stimulate active salt transport and thus raise O₂ consumption. They also increase the rate at which NO is scavenged by O₂⁻, thereby indirectly reducing O₂⁻ supply. In addition, oxidative stress correlates with a decrease in metabolic efficiency, likely due to mitochondrial uncoupling (44); that is, the rate of oxygen consumption increases without a subsequent increase in the rate of active sodium transport, manifested as a decrease in the TQ ratio. Studies in hypertensive rats (53), which operate under conditions of increased oxidative stress, have shown an approximate halving of the TQ ratio compared with their nonhypertensive counterparts. Thus we assume that a doubling of superoxide concentration (past some threshold) corresponds to a 50% reduction of the TQ ratio.

To represent these effects, we add to the model a dependence of the TQ ratio on superoxide concentration, such that above a certain threshold O₂⁻ concentration, ${C_{{{\text{O}}_2^{-}}_{}}}_{,\text{thresh}}$, the TQ ratio is assumed to decrease linearly with increasing O₂⁻ concentration. Since O₂ consumption cannot increase without bound, it is assumed that the TQ ratio reaches its minimum at one-half of the baseline TQ ratio, for O₂⁻ concentrations larger than twice the value of ${C_{{\text{O}}_2^{-}}}_{,\text{thresh}}$. In other words, the value of the TQ ratio takes the following form (see Fig. 5):

$${\text{TQ}}_i=\{\begin{array}{ll}{\text{TQ}}_{i,\text{base}}\hfill & \text{if }{C}_{i,{\text{O}}_2^{-}}<C_{i,{\text{O}}_2^{-},\text{thresh}}\hfill \\ -\frac{{\text{TQ}}_{i,\text{base}}}{\text{2}}\left(\frac{C_{i,{\text{O}}_2^{-}}}{C_{{\text{O}}_2^{-},\text{thresh}}}\right)+\frac{3{\text{TQ}}_{i,\text{base}}}{2}\hfill & \text{if }{C}_{{\text{O}}_2^{-},\text{thresh}}\le C_{i,{\text{O}}_2^{-}}\le 2C_{{\text{O}}_2^{-},\text{thresh}}\hfill \\ \frac{{\text{TQ}}_{i,\text{base}}}{2}\hfill & \text{if }{C}_{i{\text{,O}}_2^{-}}>2C_{{\text{O}}_2^{-},\text{thresh}}.\hfill \end{array}$$ (18)

where TQ_i,base is set to 18 in the ascending and descending limbs and 12 in the collecting duct, based on transport simulations for tubular epithelia (39, 50). ${C_{{\text{O}}_2^{-}}}_{,\text{thresh}}$ is set to a value assuming no oxidative stress in the base case but increased oxidative stress with a doubling of superoxide generation rate (i.e., ${C_{{\text{O}}_2^{-}}}_{,\text{thresh}}$ = 20 pM).

Fig. 5.

Assumed form of TQ ratio vs. superoxide concentration, where TQ is the number of moles of Na actively reabsorbed per mole of O₂ consumed under maximum efficiency.

To determine the most potent ways in which oxidative stress affects medullary oxygenation, we conducted simulations in the base case and with doubled maximal superoxide generation rate, with and without the dependence of TQ on O₂⁻ concentration and with and without the inclusion of [NO]-induced vasodilation. The results of these simulations are shown in Fig. 6.

Fig. 6.

Average DVR radius (μm; A), medullary O₂ supply (pmol·min⁻¹·nephron⁻¹; B), mTAL O₂ consumption (pmol·min⁻¹·nephron⁻¹; C), and average SAL Po₂ (mmHg; D, in the base case), with $G_{i,{\text{O}}_{\text{2}}^{-}}^{\max }$ × 200%, and with $G_{i,{\text{O}}_{\text{2}}^{-}}^{\max }$ × 200% and TQ dependence on [O₂⁻], both with (white bars) and without (black bars) [NO]-induced vasodilation.

When the maximal O₂⁻ generation rate is doubled but TQ is maintained constant (2nd set of bars), the increased superoxide levels (average SAL [O₂⁻] increases to 4.5 pM from 2.3 pM in the base case) lead to a slight increase in the rate of active Na⁺ transport ${\mathrm{\Psi }}_{Na}^{active}$; however, since average SAL Po₂ is less than the critical Po₂ (P_i,c in Eq. 11), the increase in ${\mathrm{\Psi }}_{Na}^{active}$ is blunted by the decrease in the fraction of active transport that is supported by aerobic metabolism, Θ(P_i,O2). In other words, an increase in anaerobic vs. aerobic metabolism acts to mitigate the effects of enhanced salt reabsorption on O₂ consumption. The net result is just a small reduction in Po₂ with increased O₂⁻ concentration.

The third set of bars accounts for TQ decreases in response to a twofold increase in O₂⁻ synthesis. Since SAL [O₂⁻] ranges from 2 to 26 pM, TQ at certain depths of the OM is reduced by 15%; in other mTALs, the maximum [O₂⁻] is over 40 pM, so that at some depths of the tubule, TQ is reduced by 50%. In this scenario, doubling the maximal O₂⁻ generation rate has a noticeable impact on O₂ consumption and on average SAL Po₂ (Fig. 6C). The model predicts that with increased oxidative stress, there is a 14% decrease in Po₂ compared with base case, without [NO]-induced vasodilation. With [NO]-induced vasodilation included, there is a 16% decrease in Po₂. While NO-induced vasodilation can help blunt the effects of increased oxidative stress (by maintaining a slightly higher Po₂), the increased superoxide levels result in a sufficiently large reduction in metabolic efficiency to significantly lower short ascending limb Po₂ in all cases.

Hyperfiltration and medullary oxygenation.

Experiments have demonstrated an increase in cell injury and hypoxia with increased filtration/flow rate (hyperfiltration), particularly prevalent in diabetes (5, 36). It has also been shown that acute hyperfiltration leads to increased superoxide production; the following equation is extrapolated from Fig. 2 in Ref. 23:

$$G_{i,{\text{O}}_2^{-}}^{\max }=G_{i,{\text{O}}_2^{-}}^{\max \ast }\left[\frac{\left(c_1\left|F_{\text{V}}^i\right|+c_2\right)}{c_{\text{scale}}}\right],$$ (19)

where $G_{i,{\text{O}}_{\text{2}}^{-}}^{{\max }^{*}}$ is the maximal rate of superoxide production when flow rate is at its baseline value (6 nl/min); F_Vⁱ is the flow rate in tubule or vessel i (in nl/min); c₁ = 1.33 (min/nl); c₂ = 10.3; and c_scale = 18.3 (23).

To assess the impact of flow-induced increases in $G_{i,{\text{O}}_{\text{2}}^{-}}^{{\max }^{*}}$, we conducted simulations of hyperfiltration by increasing descending limb inflow by 60%, with and without dependence of O₂⁻ production on flow rate, with and without dependence of TQ on O₂⁻ concentration, and with and without [NO]-induced vasodilation.

In the presence of hyperfiltration, inclusion of O₂⁻ generation dependence on flow rate produced virtually identical results to the base case, when that dependency was not included (average SAL [O₂⁻] concentration increased only slightly, from 2.27 to 2.33 pM). However, inclusion of both dependencies (O₂⁻ dependence on flow rate and TQ dependence on O₂⁻) simultaneously produced a 2.7% decrease in average SAL Po₂ with [NO]-induced vasodilatory effects included, and a 1.9% decrease with fixed DVR diameters, due to slight increases in O₂ consumption. While these decreases in Po₂ are small, these results suggest that the most likely pathway to hypoxia via hyperfiltration is from increased superoxide generation leading to a decrease in TQ; that is, the model does not predict a change in Po₂ simply from a flow-induced increase in O₂⁻ production and its subsequent effect on the active Na⁺ transport.

Effects of spatial inhomogeneity.

To determine the impact of the separation of mTAL from DVR on medullary oxygenation, we varied their degree of separation. The model represents the separation between mTAL and DVR by assigning them to different interconnected “regions.” These regions are separated by boundaries that allow diffusion of solutes. The more solute-permeable these boundaries are, the less separated the mTAL and DVR are. Thus, to vary their degree of separation, we scaled the interregion solute permeabilities by factors of ρ = 0.1, 10, and 100 (with ρ = 1 corresponding to the base case). This was done both for the OM and IM. When ρ = 0.1, there is maximal separation among the tubules and vessels; when ρ = 100, there is minimal separation.

In Fig. 7, the O₂ supply to the OM and deep IM (2.5 mm below the OM/IM boundary) is shown, for varying values of ρ, with and without NO-induced vasodilation. With NO-induced vasodilation, the O₂ supply to the OM increases with decreasing separation, as NO diffuses from R3 and R4 (which produce the most NO) to the DVR in R1 and R2. In the deep IM, as separation decreases, O₂ diffuses away from high-O₂ areas such as the DVR; that is, increasing ρ causes O₂ to diffuse in the opposite direction of NO. As can be seen in the black bars of Fig. 7B, this leads to a decrease in O₂ supply to the deep IM for the fixed DVR radius case. In the NO-induced vasodilation case, however, the diffusion of NO into the DVR leads to vasodilation, which causes O₂ supply to the deep IM to increase. These competing effects lead to nonmonotonic behavior in the NO vasodilation case as ρ increases (see Fig. 7B, white bars). Below a certain permeability threshold, reducing the separation between regions enhances the delivery of O₂ to the deep IM; above that threshold, it has the opposite effect.

Fig. 7.

O₂ supply (pmol·min⁻¹·nephron⁻¹) to the OM (A) and deep IM (2.5 mm below OM/IM boundary) (B) under different degrees of regionalization. Cases shown: ρ = 0.1, base case, ρ = 10, and ρ = 100, both with (white bars) and without (black bars) [NO]-induced vasodilation.

In Fig. 8, total medullary O₂ consumption, as well as O₂ consumption in the OM and IM individually, are shown, for varying values of ρ and with and without NO-induced vasodilation. In the OM, total consumption decreases with decreasing separation (Fig. 8B), as higher [NO] and lower [Na⁺] levels in the mTALs and OMCD reduce active transport. In the IM, changes in O₂ consumption are correlated with changes in O₂ delivery (compare Figs. 7B and 8C). At the papillary tip, Po₂ steadily decreases as separation decreases in the fixed DVR radius case (from 6.0 mmHg with ρ = 0.1 to 4.2 mmHg with ρ = 100). In the NO-induced vasodilation case, however, the effects of changing ρ on papillary Po₂ are nonmonotonic, similar to the effects on O₂ delivery and consumption. Po₂ at the papilla is predicted to be 5.2 mmHg with ρ = 100, 5.3 mmHg with ρ = 10, and 4.6 mmHg in the base case. With maximal separation (ρ = 0.1), the large decrease in O₂ supply leads to a large decrease in Po₂, to 3.2 mmHg.

Fig. 8.

O₂ consumption (pmol·min⁻¹·nephron⁻¹) in the whole medulla (A), in the OM (B), and in the IM (C), under different degrees of regionalization. Cases shown: ρ = 0.1, base case, ρ = 10, and ρ = 100, both with (white bars) and without (black bars) [NO]-induced vasodilation.

Figures 9 and 10 illustrate the effects of a decrease in O₂ supply via a decrease in DVR inflow, from 8 nl/min in the base case to 6 nl/min. Overall trends are similar to the ones observed with the base case DVR inflow of 8 nl/min. That is, the delivery of O₂ to the deep IM decreases with decreasing separation between regions assuming no NO-induced vasodilation, but it varies in a nonmonotonic manner when NO-induced vasodilation is accounted for in Fig. 9. Furthermore, O₂ consumption in the deep IM varies in tandem with O₂ supply to that region (Fig. 10). In addition, papillary Po₂ decreases with decreasing separation in the fixed DVR radius case (from 4.5 mmHg with ρ = 0.1 to 3.2 mmHg with ρ = 100) but changes nonmonotonically in the NO-induced vasodilation case, with Po₂ levels at 2.9, 3.5, 4.5, and 4.3 mmHg when ρ = 0.1, 1, 10, and 100. Note that the absolute differences between the cases with and without NO-induced vasodilation are the same at both flow rates; that is, the decrease in DVR inflow accentuates the relative differences between the NO vasodilation and fixed resistance cases.

Fig. 9.

O₂ supply (pmol·min⁻¹·nephron⁻¹) to the OM (A) and deep IM (2.5 mm below OM/IM boundary) (B) with a DVR inflow rate of 6 nl/min (base case is 8 nl/min), under different degrees of regionalization. Cases shown: ρ = 0.1, Base Case, ρ = 10, and ρ = 100, both with (white bars) and without (black bars) [NO]-induced vasodilation.

Fig. 10.

O₂ consumption (pmol·min⁻¹·nephron⁻¹) in the whole medulla (A), in the outer medulla (B), and in the inner medulla (C) with a DVR inflow rate of 6 nl/min (base case is 8 nl/min), under different degrees of regionalization. Cases shown: ρ = 0.1, base case, ρ = 10, and ρ = 100, both with (white bars) and without (black bars) [NO]-induced vasodilation.

DISCUSSION

We extended a detailed mathematical model of nitric oxide and superoxide transport in the rat renal medulla (17) to include the effects of NO-mediated vasodilation. The model represents the radial organization of renal medullary vessels and tubules, centered around vascular bundles in the OM and around CDs in the IM. The model predicts 1) concentrations of Na⁺, urea, O₂, NO, and O₂⁻ in all represented structures, as well as Hb, HbO₂, and HbNO in the RBCs, and 2) flow rates of water and solutes in the vessels and tubules. Simulation results suggest that the two ways in which NO acts to prevent medullary hypoxia, namely by increasing O₂ supply via vasodilation and by reducing active sodium transport and therefore O₂ consumption, have a comparable impact on medullary oxygenation. Furthermore, model results imply that a reduction in tubular transport efficiency is the main factor by which increased O₂⁻ levels lead to hypoxia and that flow-induced increases in O₂⁻ levels during hyperfiltration are not a likely pathway to medullary hypoxia.

Contribution of NO to maintenance of medullary Po₂.

Experiments have demonstrated that NO helps to maintain Po₂ levels and blunt hypoxia (6). This may be achieved via the capacity of NO to dilate DVR, increasing blood flow and thus O₂ supply, and/or via the ability of NO to inhibit active sodium transport and thus lower O₂ consumption. These two NO-mediated pathways are difficult to modulate individually in a laboratory experiment, and the assessment of their relative contributions to medullary oxygenation is a question amenable to our modeling approach.

Model results suggest that the NO-induced decrease in O₂ consumption and increase in O₂ supply contribute to raising medullary Po₂ to a similar extent. In other words, the vasodilatory effects of NO per se significantly increase medullary oxygenation when the generation rate of NO is raised (Fig. 4). Conversely, when NO levels are reduced, O₂ supply to the medulla is substantially reduced, and the predicted mTAL Po₂ is lower than when the vasoactive properties of NO are not considered (Fig. 4D). The model thus predicts that the vasoactive properties of NO may exacerbate hypoxic conditions when NO is reduced.

Effects of oxidative stress on medullary Po₂.

Elevated oxidative stress in the rat kidney leads to mitochondrial uncoupling, which reduces tubular transport efficiency and thus increases O₂ consumption (44). This is manifested as a decrease in the TQ ratio, whereby the number of moles of O₂ that are consumed to actively transport one mole of Na⁺ increases. However, increased O₂⁻ levels also stimulate active sodium transport, so it is not clear whether the decrease in Po₂ that accompanies oxidative stress is due to more active pumping or to a reduction in pumping efficiency. Furthermore, there are multiple sources of oxidative stress in the kidney. In particular, experiments have demonstrated a dependence of O₂⁻ generation on flow rate, such that hyperfiltration leads to increased O₂⁻. We thus investigated which of these pathways contributes most to oxidative stress leading to hypoxia and whether the increase in O₂⁻ levels from hyperfiltration was sufficient to reduce medullary oxygenation.

Model results suggest that increased O₂⁻ levels, whether from hyperfiltration or other sources of oxidative stress, are not sufficient to significantly lower Po₂, unless the effects of O₂⁻ on TQ are considered. Even though increases in O₂⁻ levels elevate the rate of active Na⁺ transport, the compensating effect of an increased ratio of anaerobic-to-aerobic metabolism serves to mitigate increases in O₂ consumption and thus prevent large decreases in Po₂. In the case where the impact of O₂⁻ on TQ is included, the model predicts more noticeable decreases in Po₂ with increases in O₂⁻ (Fig. 6D). This indicates that the effects of O₂⁻ on Po₂ mostly stem from a reduction in metabolic efficiency, rather than from an increase in ${\mathrm{\Psi }}_{Na}^{active}$. In addition, model results suggest that in the physiologically relevant range of hyperfiltration, the resulting flow-induced increase in O₂⁻ levels is only ∼25%. If one assumes that “oxidative stress” occurs when medullary O₂⁻ levels increase by at least a factor of 2, then the model predicts that oxidative stress does not result from hyperfiltration per se. In other words, some other source of oxidative stress, leading to mitochondrial uncoupling and lowered metabolic efficiency, is a more likely pathway to hypoxia.

Impact of structural organization.

We previously examined the impact of the three-dimensional organization of tubules and vessels in the renal medulla on O₂ transport with a model that did not include NO and O₂⁻ (18). This previous model predicted that in the absence of distinct interstitial regions, oxygen delivery to the IM would drastically decrease, with the terminal IM nearly completely deprived of oxygen. Based on these results, we postulated that one of the functional roles of the three-dimensional medullary architecture may be to preserve oxygen delivery to the papilla. The present simulations suggest a more nuanced picture. In the IM, NO and O₂ travel radially in opposite directions: NO diffuses into the DVR thereby favoring vasodilation and a higher O₂ supply, whereas O₂ diffuses away from blood vessels. When the degree of separation between regions is reduced, that is, when the interregion permeability increases but remains below a certain threshold, these counterbalancing effects result in a higher delivery of O₂ to the deep IM. As the interregion permeability is elevated beyond that threshold, O₂ supply to the deep IM starts to decline. In other words, a perfectly homogeneous interstitium would be detrimental to the oxygenation of the deep medulla. However, the degree of separation which we assume represents physiological conditions may not maximize O₂ delivery to the IM.

Interstitial pH varies within the medulla, with the IM being more acidic than the OM. Radially, interstitial fluid surrounding the thick ascending limbs has been shown to have a pH lower than that in the vascular bundles (8); similarly, interstitial fluid surrounding the IMCD is more acidic than that around the DVR (28). In a previous study (18), we assessed the impact of varying pH on medullary oxygenation and sodium transport. To that end, we allowed the O₂ dissociation curve to shift to the right under conditions of increased acidity. Modeling results suggest that the acidity in the interbundle and intrabundle regions facilitates the dissociation of HbO₂, leading to higher O₂ availability and a slight increase in active Na⁺ reabsorption from the TAL and IMCD. Our simulations suggest that the effects of pH on NO-mediated vasodilation and oxidative stress are insignificant (results not shown). Thus the pH effects were not included in the present model.

Model limitations and future extensions.

One major limitation of the model stems from the lack of experimental measurements of NO and O₂⁻ concentrations in the renal medulla with which to compare our results. To the best of our knowledge, absolute measurements of medullary O₂⁻ concentrations have not been made, and measurements of NO in the medulla vary over an order of magnitude. Equations that describe the effects of NO and O₂⁻ on active Na⁺ transport (Eqs. 12 and 13) involve parameters that were fitted to available experimental data; however, due to the dearth of such data in the medulla, the parameter choices are necessarily uncertain. In addition, the assumed form of Eq. 3 that describes how DVR radius varies with changing NO concentrations is extrapolated from a small number of data points (25). In light of these issues, the model would benefit from experimental measurements of O₂⁻ levels in the medulla, as well as sodium transport in differing NO and O₂⁻ environments, and DVR radii in a range of NO concentrations.

Fluid dynamics in the vasa recta is represented as quasi steady-state Poiseuille flow, which assumes laminar flow through a long pipe with uniform radius. Because the RBC radius is comparable to that of DVR lumen, the assumption of laminar flow may not be appropriate, and the conditions for Poiseuille flow are at best approximately satisfied. A more realistic model would be based on fluid-structure interactions, with fluid dynamics described by the Navier-Stokes equations and with the RBCs modeled explicitly as deformable objects immersed in the fluid. However, the computations required for solving the Navier-Stokes equations and the fluid-structure interaction equations are very time-consuming and beyond the scope of this model.

Avoidance of medullary hypoxia requires the interplay of a variety of interrelated mechanisms, affecting oxygen delivery and consumption. In particular, the balance between the opposing solutes NO and O₂⁻ helps to maintain homeostatic levels of medullary oxygenation. Future modeling studies investigating the impairment of autoregulatory mechanisms may provide further insight into the causes of hypoxic conditions in disease states such as diabetes and hypertension.

GRANTS

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

DISCLOSURES

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

AUTHOR CONTRIBUTIONS

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