A computational model for simulating solute transport and oxygen consumption along the nephrons

Abstract

The goal of this study was to investigate water and solute transport, with a focus on sodium transport (T_Na) and metabolism along individual nephron segments under differing physiological and pathophysiological conditions. To accomplish this goal, we developed a computational model of solute transport and oxygen consumption (Q_O2) along different nephron populations of a rat kidney. The model represents detailed epithelial and paracellular transport processes along both the superficial and juxtamedullary nephrons, with the loop of Henle of each model nephron extending to differing depths of the inner medulla. We used the model to assess how changes in T_Na may alter Q_O2 in different nephron segments and how shifting the T_Na sites alters overall kidney Q_O2. Under baseline conditions, the model predicted a whole kidney T_Na/Q_O2, which denotes the number of moles of Na⁺ reabsorbed per moles of O₂ consumed, of ∼15, with T_Na efficiency predicted to be significantly greater in cortical nephron segments than in medullary segments. The T_Na/Q_O2 ratio was generally similar among the superficial and juxtamedullary nephron segments, except for the proximal tubule, where T_Na/Q_O2 was ∼20% higher in superficial nephrons, due to the larger luminal flow along the juxtamedullary proximal tubules and the resulting higher, flow-induced transcellular transport. Moreover, the model predicted that an increase in single-nephron glomerular filtration rate does not significantly affect T_Na/Q_O2 in the proximal tubules but generally increases T_Na/Q_O2 along downstream segments. The latter result can be attributed to the generally higher luminal [Na⁺], which raises paracellular T_Na. Consequently, vulnerable medullary segments, such as the S3 segment and medullary thick ascending limb, may be relatively protected from flow-induced increases in Q_O2 under pathophysiological conditions.

in the united states, 1 in 10 adults, or more than 20 million, have some level of chronic kidney disease (3). While the etiology of chronic kidney diseases remains incompletely understood despite substantial clinical and physiological research, renal hypoxia is believed to play a key role (9, 10, 12). Renal hypoxia can be caused by pathophysiological conditions, such as diabetes or hypertension, that induce mismatched changes in renal oxygen delivery and/or oxygen consumption (Q_O2) (8).

Relative to its size, the kidneys receive a large amount of blood: 20–25% of cardiac output. A portion of renal plasma flow is filtered (in humans, the glomerular filtration rate is ∼180 l/day) and enters the nephron; most of that filtrate (∼99% of filtered NaCl and fluid) is subsequently reabsorbed. To support that reabsorptive process, the kidneys require a large amount of energy. Indeed, although the human kidneys constitute only 0.5% of body mass, they consume 10% of the oxygen used in cellular respiration. Much of the oxygen consumed by the kidneys is used to support the active reabsorption of Na⁺, primarily along the proximal tubule and thick ascending limb.

Efficiency of Na⁺ reabsorption varies greatly along the nephron (16). Thus the primary goal of this study and of the companion study (19) is to investigate how physiological and pathophysiological changes in Na⁺ transport (T_Na) alter Q_O2 in different segments and how shifting the T_Na site alters renal Q_O2. Distal segments are known to require substantially more O₂ than the proximal tubule to reabsorb the same amount of Na⁺ (16). Thus, if proximal tubule transport is compromised, the resulting shift of T_Na to more distal sites within the nephron should yield a major increase in total renal Q_O2.

In a recent study (21), we developed a detailed epithelial cell-based computational model of solute and water transport along the superficial nephron of a rat kidney. In this study, we have extended that model to include a population of juxtamedullary nephrons, and examine the effects of varying single-nephron glomerular filtration rate (SNGFR) on solute transport and renal metabolism. In the companion study (19), the model is applied to investigate the extent to which inhibitors of T_Na along the nephron, such as loop diuretics, impact Q_O2 and T_Na efficiency and how those effects may differ among superficial and juxtamedullary nephrons.

MATHEMATICAL MODEL

We have developed an epithelial cell-based model of solute transport and Q_O2 along different nephron populations of a rat kidney. The model represents six classes of nephrons: a superficial nephron (denoted “SF”) and five juxtamedullary nephrons. The latter include loops of Henle that reach into differing depths (1, 2, 3, 4, and 5 mm) of the inner medulla, taken to be 5-mm long; we refer to these nephrons as “JM-1,” “JM-2,” “JM-3,” “JM-4,” and “JM- 5,” respectively. Based on anatomic findings (17), we assume that the superficial nephrons account for 2/3 of the nephron population, with the remainder being juxtamedullary; for simplicity, mid-cortical nephrons are not included. Most of the long loops turn within the upper inner medulla (17). We assume that the number ratios (per nephron) of the six nephron classes are n_SF = 2/3, n_JM−1 = 0.4/3, n_JM−2 = 0.3/3, n_JM−3 = 0.15/3, n_JM−4 = 0.1/3, and n_JM−5 = 0.05/3.

The superficial nephron model is based on the previously applied model (21), which extends from the Bowman's capsule to the papillary tip and includes the proximal tubule, short descending limb, thick ascending limb, distal convoluted tubule, connecting tubule, and the collecting duct. The juxtamedullary nephron models include longer descending limbs and, in addition, ascending thin limbs, i.e., the inner-medullary segments of the loops (see below). Each nephron segment is represented as a tubule lined by a layer of epithelial cells, with apical and basolateral transporters that vary according to cell type (Fig. 1). Measurements in rats have indicated that the SNGFR of juxtamedullary nephrons is 1.5–2 times that of superficial ones (11, 13, 15, 27). Thus SNGFR is set to ∼30 and ∼45 nl/min for the superficial and juxtamedullary nephrons, respectively (more below). The transport capacity of the juxtamedullary proximal tubule is assumed to be 75% higher than that of the superficial proximal tubule, such that fluid flows into the S3 segments are predicted to be between 11 (for the superficial nephron) and 13 nl/min. The assumption of the relative transport capacity of the two proximal tubule populations is consistent with perfused tubule studies in the rabbit (14).

Fig. 1.

Schematic diagram of the nephron system (not to scale). The model includes 1 representative superficial nephron and 5 representative juxtamedullary nephrons, each scaled by the appropriate population ratio. Only the superficial nephron and one juxtamedullary nephron are shown. Along each nephron, the model accounts for the transport of water and 15 solutes (see text). The distal tubule is divided into DCT1 and DCT2 (see text). The diagram displays only the main Na⁺, K⁺, and Cl⁻ transporters. mTAL, medullary thick ascending limb; cTAL, cortical thick ascending limb; DCT, distal convoluted tubule; PCT, proximal concoluted tubule; CNT, connecting duct; CCD, cortical collecting duct; SDL, short or outer-medullary descending limb; LDL/LAL, thin descending/ascending limb; OMCD, outer-medullary collecting duct; IMCD, inner-medullary collecting duct.

The model accounts for 15 solutes: Na⁺, K⁺, Cl⁻, HCO₃⁻, H₂CO₃, CO₂, NH₃, NH₄⁺, HPO₄²⁻, H₂PO₄⁻, H⁺, HCO₂⁻, H₂CO₂, urea, and glucose. The model is formulated for steady state and predicts luminal fluid flow, hydrostatic pressure, luminal fluid solute concentrations, and, with the exception of the descending limb and thin ascending limb segments, cytosolic solute concentrations, membrane potential, and transcellular and paracellular fluxes.

Descending and ascending thin limbs.

For the superficial nephron, the S3 segment and the thick ascending limb are connected by the short descending limb segment; for the juxtamedullary nephrons, they are connected by the descending and ascending thin limbs. Because the transport characteristics of these thin limb segments are not well characterized, simple, single-barrier transport models are used (see Table 1). The juxtamedullary descending thin limbs are assumed to consist of functionally distinct segments: the outer-medullary segment and the initial 40% of the inner-medullary segment are assumed to be moderately water permeable and impermeable to Na⁺, K⁺, Cl⁻, and urea, whereas the terminal 60% of the inner-medullary segment is water impermeable (23, 28, 35); highly permeable to Na⁺, Cl⁻, and urea (23, 28); and moderately permeable to K⁺. The ascending thin limbs are assumed to be highly permeable to Na⁺, Cl⁻, and urea; moderately permeable to K⁺; and impermeable to water. Transport parameters of loop segments are given in Table 1. In these single-barrier transport models, the intracellular concentrations and membrane potential of the descending limb epithelial cells are not computed (unlike in other segments); only changes in luminal volume flow and pressure are tracked. The luminal membrane areas of tubular cells are set to 2.0 cm² per cm² epithelial area, as in thick ascending limb cells. The surface areas of the tight junction and the interspace-basal interface are set to 0.001 and 0.02 cm² per cm² epithelial area, as in all other segments.

Table 1.

Loop of Henle segmental transport parameters

Permeability	SDL	LDL	LAL
Water, μm/s	2,200 | 0	2,200 | 0	0
Na+, ×10−5 cm/s	1	0 | 80	80
K+, ×10−5 cm/s	1	0 | 100	100
Cl−, ×10−5 cm/s	1	0 | 80	80
HCO3−, ×10−5 cm/s	1	0 | 20	20
H2CO3, ×10−5 cm/s	65	65	65
CO2, ×10−5 cm/s	1,200	1,200	1,200
HPO42−, ×10−5 cm/s	0	0	0
H2PO4−, ×10−5 cm/s	0	0	0
NH3, ×10−5 cm/s	85	85	85
NH4+, ×10−5 cm/s	1	0 | 20	20
H+, ×10−5 cm/s	850	850	850
HCO2−, ×10−5 cm/s	0	0	0
H2CO2, ×10−5 cm/s	0	0	0
Glucose, ×10−5 cm/s	0	0	0
Urea, ×10−5 cm/s	5	80	80

Vertical line separates transition of initial and terminal descending limb transport properties (see text). SDL, short or outer-medullary descending limb; LDL and LAL, inner-medullary segments of long descending and ascending limbs.

Tubule coalescence.

The model assumes that each class of connecting tubules coalesces successively, reducing the total luminal cross-sectional area. That convergence is described by ω_CNT(x), which denotes the fraction of connecting tubules remaining at coordinate x_CNT

$${\omega }_{\text{CNT}}(x_{\text{CNT}})=2^{-2.32\hspace{0.17em}{x}_{\text{CNT}}\hspace{0.17em}/{\hspace{0.17em}L}_{\text{CNT}}},$$ (1)

where L_CNT denotes the connecting tubule length and x_CNT denotes the distance from the connecting tubule entrance. At the end of the connecting tubule, 20% of the population remains [i.e., ω_CNT(L_CNT) = 0.2]. These remaining connecting tubules converge into the cortical collecting duct.

The inner-medullary collecting ducts coalesce in a manner similar to the connecting tubules. Let ω_IMCD(x_IMCD) denote the number of inner-medullary collecting ducts per nephron:

$${\omega }_{\text{IMCD}}(x_{\text{IMCD}})=0.2\times \left(1-0.95{\left(\frac{x_{\text{IMCD}}}{L_{\text{IMCD}}}\right)}^2\right)\hspace{0.17em}{\exp }^{-2.75\hspace{0.17em}{x}_{\text{IMCD}}\hspace{0.17em}/\hspace{0.17em}{L}_{\text{IMCD}}},$$ (2)

where L_IMCD denotes the inner-medullary collecting duct length and x_IMCD denotes the distance from its entrance. If we assume that there are 36,000 nephrons per kidney, then at the papillary tip (i.e., x_IMCD = L_IMCD), 23 collecting ducts remain.

Tubular conservation and fluid pressure equations.

Within the lumen of a noncoalescing tubule i, conservation of water and nonreacting solutes is given by

$$\frac{d{Q}_i}{dx}=J_{v,i}$$ (3)

$$\frac{d}{dx}(Q_i{C}_{k,i})=J_{k,i}$$ (4)

where Q_i denotes volume flow (per tubule) and C_k,i denotes the concentration of solute k. J_v,i denotes overall (transepithelial and paracellular) water flux and J_k,i denotes the analogous solute flux.

For the reacting solutes, conservation is applied to the total buffers (36):

$$\frac{d}{dx}(Q_i{C}_{{\text{CO}}_2,i}+Q_i{C}_{{\text{HCO}}_3,i}+Q_i{C}_{{\text{H}}_2{\text{CO}}_3,i})=J_{{\text{CO}}_2,i}+J_{{\text{HCO}}_3,i}+J_{{\text{H}}_2{\text{CO}}_3,i}$$ (5)

$$\frac{d}{dx}(Q_i{C}_{{\text{HPO}}_4,i}+Q_i{C}_{{\text{H}}_2{\text{PO}}_4,i})=J_{{\text{HPO}}_4,i}+J_{{\text{H}}_2{\text{PO}}_4,i}$$ (6)

$$\frac{d}{dx}(Q_i{C}_{{\text{NH}}_3,i}+Q_i{C}_{{\text{NH}}_4,i})=J_{{\text{NH}}_3,i}+J_{{\text{NH}}_4,i}$$ (7)

$$\frac{d}{dx}(Q_i{C}_{{\text{HCO}}_2,i}+Q_i{C}_{{\text{H}}_2{\text{CO}}_2,i})=J_{{\text{HCO}}_2,i}+J_{{\text{H}}_2{\text{CO}}_2,i}$$ (8)

The buffer pairs are assumed to be in equilibrium:

$$\text{pH}={\text{pK}}_A-\log \frac{C_{A,i}}{C_{B,i}}$$ (9)

where the buffer pairs (A,B) are (HCO₃⁻,H₂CO₃), (HPO₄²⁻,H₂PO₄⁻), (NH₃,NH₄⁺), and (HCO₂⁻,H₂CO₂). pH is given by conservation of hydrogen ion:

$${\displaystyle \sum _k\left(\frac{d}{dx}{Q}_i{C}_{k,i}\right)}={\displaystyle \sum _k{J}_{k,i}}$$ (10)

where the summation index k is applied over the solutes H⁺, NH₄⁺, H₂PO₄⁻, H₂CO₃, and H₂CO₂.

For a coalescing tubule (i = CNT or IMCD), the equations are modified by scaling water and solute flows by the tubule population ω_i. For example, conservation of luminal fluid and nonreacting solutes is given by

$$\frac{d}{dx}({\omega }_i{Q}_i)=J_{v,i}$$ (11)

$$\frac{d}{dx}({\omega }_i{Q}_i{C}_{k,i})=J_{k,i}$$ (12)

Equations 5–8 are modified similarly.

At the entrance of the cortical collecting duct (CCD), all types of connecting tubules converge. This configuration yields the following cortical collecting duct inflow conditions. For luminal fluid and nonreacting solutes:

$$0.2\times Q_{\text{CCD}}=n_{\text{SF}}{Q}_{\text{CNT}}^{\text{SF}}+{\displaystyle \sum _l{n}_{\text{JM}-l}{Q}_{\text{CNT}}^{\text{JM}-l}},$$ (13)

$$0.2\times Q_{\text{CCD}}{C}_{k,\text{CCD}}=n_{\text{SF}}{Q}_{\text{CNT}}^{\text{SF}}{C}_{k,\text{CNT}}+{\displaystyle \sum _l{n}_{\text{JM}-l}{Q}_{\text{CNT}}^{\text{JM}-l}{C}_{k,\text{CNT}}^{\text{JM}-l}},$$ (14)

The factor 0.2 takes into account the convergence of five connecting tubules into one cortical collecting duct. For the reacting solutes, total buffer is conserved:

$$0.2\times Q_{\text{CCD}}{\displaystyle \sum _k{C}_{k,\text{CCD}}}=n_{\text{SF}}{Q}_{\text{CNT}}^{\text{SF}}{\displaystyle \sum _k{C}_{k,\text{CNT}}^{\text{SF}}}+{\displaystyle \sum _l\left(n_{\text{JM}-l}{Q}_{\text{CNT}}^{\text{JM}-l}{\displaystyle \sum _k{C}_{k,\text{CNT}}^{\text{JM}-l}}\right)},$$ (15)

The summation over k is applied to each buffer group. For example, one instance of Eq. 15 is applied over CO₂, HCO₃⁻, and H₂CO₃, another over HPO₄²⁻ and H₂PO₄⁻, and so forth. Each buffer pair is assumed to be in equilibrium and satisfies Eq. 9. Conservation of hydrogen ion is also applied:

$$0.2\times Q_{\text{CCD}}{\displaystyle \sum _k{C}_{k,\text{CCD}}}=n_{\text{SF}}{Q}_{\text{CNT}}^{\text{SF}}{\displaystyle \sum _k{C}_{k,\text{CNT}}^{\text{SF}}}+{\displaystyle \sum _l\left(n_{\text{JM}-l}{Q}_{\text{CNT}}^{\text{JM}-l}{\displaystyle \sum _k{C}_{k,\text{CNT}}^{\text{JM}-l}}\right)}$$ (16)

where the summation index k is applied over the solutes H⁺, NH₄⁺, H₂PO₄⁻, H₂CO₃, and H₂CO₂.

The functions in Eqs. 13, 14, 15, and 16 are evaluated at the end of the connecting tubules, x = L_CNT. Full carbonic anhydrase activity is assumed so that there is full equilibration of CO₂ and H₂CO₃. The convergence of the six nephron classes requires that their tubular fluid pressure values match at x = L_CNT. This condition is satisfied by setting, via an iterative procedure, the appropriate inflow pressure and SNGFR at the entrance of each proximal tubule; see boundary conditions below.

Tubular fluid flow is described by the pressure-driven Poiseuille flow. Along a noncoalescing tubule i, the hydrostatic pressure in the lumen, P_i, is related to volume flow Q_i (per tubule) and luminal radius r_i by

$$\frac{d{P}_i}{dx}=-\frac{8\mu Q_i}{\pi r_i^4}$$ (17)

where μ is the luminal fluid viscosity (taken as 6.4 × 10⁻⁶ mmHg/s). As the connecting tubules and inner-medullary collecting ducts coalesce, fluid flow in a given tubule increases, even as water is reabsorbed. To take into account the increase in drag resistance due to tubular coalescence, fluid viscosity along these segments is increased by 2.

Flow-dependent tubular transport.

Proximal tubule reabsorption varies proportionally to SNGFR (32). To model flow-dependent transepithelial transport, we follow the approach of Weinstein et al. (40). Specifically, the proximal tubule is assumed to be compliant, with luminal radius given by

$$r_{\text{PT}}=r_{\text{PT}}^0\hspace{0.17em}(1+{\mu }_{\text{PT}}\hspace{0.17em}(P_{\text{PT}}-P_{\text{PT}}^0))$$ (18)

where the reference radius r_PT⁰ is taken as 11.2 μm, the reference pressure P_PT⁰ is taken as 9 mmHg, and μ_PT, which characterizes tubular compliance, is set to 0.03.

To account for the modulation of transporter density by luminal flow, we determine the microvillous torque as

$${\tau }_{\text{PT}}=\frac{8\mu Q_{\text{PT}}{l}_{\text{PT,mv}}}{r_{\text{PT}}^2}\left(1+\frac{l_{\text{PT,mv}}+{\delta }_{\text{PT,mv}}}{r_{\text{PT}}}+\frac{l_{\text{PT,mv}}^2}{2r_{\text{PT}}^2}\right)$$ (19)

where l_PT,mv = 2.5 μm is the microvillous length and δ_PT,mv = 0.15 μm denotes the height above the microvillous tip where drag is considered (6). The density of apical and basolateral transporters in proximal tubule cells is scaled by:

$$1+s\left(\frac{{\tau }_{\text{PT}}}{{\tau }_{\text{PT}}^{\text{0}}}-1\right)$$ (20)

where the reference torque τ_PT⁰ is evaluated at the reference flow set to the inflow of the proximal tubule (i.e., ∼30 and 45 nl/min, respectively, for the superficial and juxtamedullary proximal tubules) and reference radius r_PT⁰ = 12.5 μm. The scaling factor s is taken to be 1.5 for the S1-S2 segment and 0.75 for the S3 segment.

The connecting tubules and cortical collecting ducts are also known to exhibit flow- dependent transepithelial transport (31, 18, 43). We adopt a simple representation of that flow dependence by scaling apical Na⁺ permeability in the connecting tubule and cortical collecting duct, respectively, by τ_CNT and τ_CCD, given by

$${\tau }_{\text{CNT}}=1+3\left(\frac{Q_{\text{CNT}}}{Q_{\text{CNT}}^{\text{0}}}-1\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_{\text{CCD}}=1+3\left(\frac{Q_{\text{CCD}}}{Q_{\text{CCD}}^{\text{0}}}-1\right)$$ (21)

where the reference flows Q_CNT⁰ and Q_CCD⁰ are taken to be 1.7 and 1.3 nl/min, respectively. Model simulations suggest that scaling apical Na⁺ permeability also induces flow dependence in K⁺ transport, consistent with experimental observations (31, 18, 43). When cortical collecting duct inflow was varied fourfold, from 65 to 260% of baseline flow, and with inflow [Na⁺], [K⁺], and [Cl⁻] adjusted so that they more closely approximately experimental conditions reported in Ref. 43, the model predicted that Na⁺ reabsorption and K⁺ secretion increased 2.4 and 2.8 times, respectively; these increases are comparable to the values reported in Ref. 43 (∼2.7 and 3.0 times as shown in Figs. 1 and 2). Flow-induced transport may well be present in other segments, e.g., the thick ascending limb. However, in the absence of adequate experimental data, we assume that tubular transport along other segments is independent of flow.

Fig. 2.

Total delivery of key solutes (A–H) and fluid (I) to the beginning of individual nephron segments, given per kidney. Juxtamedullary values are computed as weighed totals of the five representative model juxtamedullary nephrons. The model assumes a superficial-to-juxtamedullary nephron ratio of 2:1, thus, the superficial delivery values are generally higher. In each panel, the two bars for “urine” are identical since the superficial and juxtamedullary nephrons have merged at the cortical collecting duct entrance. PT, proximal tubule; DL, descending limb; mTAL, medullary thick ascending limb; DCT, distal convoluted tubule; CNT, connecting duct; CCD, cortical collecting duct. Insets: reproductions of distal segment values.

Oxygen consumption along the nephron.

Following our previous approach (20), Q_O2 is divided into two parts: the active component (Q_O2^active) provides the energy needed to actively reabsorb Na⁺, and the basal component (Q_O2^basal) supplies the energy for other transport processes and intracellular biochemical reactions. Note that Q_O2^basal includes the contributions of H⁺-ATPase and H⁺-K⁺-ATPase, which, to a large extent, are not directly coupled to Na⁺ transport. Q_O2^active is calculated based on the ATP consumption of basolateral Na⁺-K⁺-ATPase pumps. Since 1 mol of ATP is required to pump out 3 mol of Na⁺ via the pump, and oxidative metabolism yields about 5 mol of ATP per mol of O₂ consumed (30), Q_O2^active is determined as

$${\mathrm{Q}}_{\text{O2}}^{\text{active}}={\mathrm{T}}_{\text{Na}}^{\text{active}}/15$$ (22)

where T_Na^active is the rate of Na⁺ transport across Na⁺-K⁺-ATPase pumps.

In rats, the whole kidney basal-to-total Q_O2 ratio has been estimated as 25–30% (41). To the best of our knowledge, that ratio has not been determined in individual nephron segments. We assume that for the entire kidney Q_O2^basal is fixed and equal to 25% of (total) Q_O2 under baseline conditions, such that

$${\mathrm{Q}}_{\text{O2}}^{\text{basal}}=0.25({\mathrm{Q}}_{\text{O2}}^{\text{basal}}+{\mathrm{Q}}_{\text{O2}}^{\text{active*}})=(0.25/0.75){\mathrm{Q}}_{\text{O2}}^{\text{active*}}$$ (23)

where the asterisk denotes base-case conditions. Thus Q_O2^basal remains constant in all simulations. We further assume that the Q_O2^basal rate per unit length is the same for all nephron segments. With these assumptions, Q_O2^basal for a given segment is proportional to its length.

The efficiency of oxygen utilization can be evaluated by computing the number of moles of Na⁺ reabsorbed per mole of O₂ consumed:

$${\mathrm{T}}_{\text{Na}}^{\text{total}}/{\mathrm{Q}}_{\text{O2}}^{\text{total}}=\frac{{\mathrm{T}}_{\text{Na}}^{\text{active}}+{\mathrm{T}}_{\text{Na}}^{\text{passive}}}{({\mathrm{T}}_{\text{Na}}^{\text{active}}/15+{\mathrm{Q}}_{\text{O2}}^{\text{basal}})}$$ (24)

where T_Na^passive denotes the rate of passive Na⁺ reabsorption.

Other model assumptions.

The superficial nephron is divided into several functionally distinct segments. The length of the cortical (S1-S2) and outer medullary (S3) segments of its proximal tubule is taken as 0.97 and 0.13 cm, respectively. The short descending limb extends from the junction between the outer and inner stripes of the outer medulla (0.06 cm below the cortico-medullary junction) to the boundary between the outer and inner medulla (0.2 cm below the cortico-medullary junction). The initial 40% of the short descending limb is assumed to be highly water permeable, whereas the remainder is water impermeable (35). The medullary and cortical thick ascending limbs are each taken to be 0.2-cm long; the distal convoluted tubule is 0.1-cm long; the connecting tubule is 0.2-cm long.

The juxtamedullary nephron is similarly divided into several segments. The lengths of its S1, S2, S3, and outer-medullary descending limb segments are the same as that of the superficial nephron. Different juxtamedullary nephrons reach into differing inner-medullary depths; the inner-medullary length of its descending and ascending thin limbs is taken to be 0.1, 0.2, 0.3, 0.4, or 0.5 cm. The upper 40% of the descending limb is assumed to be functionally distinct from the terminal segment (see above). The medullary and cortical thick ascending limbs are each taken to be 0.2- and 0.05-cm long, respectively; the distal convoluted tubule is 0.1-cm long; the connecting tubule is 0.3-cm long.

The superficial and juxtamedullary nephrons merge at the entrance of the cortical collecting duct. The cortical and outer medullary collecting ducts are each 0.2-cm long; and the inner medullary collecting duct is 0.5-cm long.

The distal tubule is assumed to consist of two functionally distinct segments. Along the initial 2/3 of the segment (DCT1), the sodium-chloride cotransporter NCC is expressed homogeneously, and the epithelial sodium channel ENaC is not present. The remainder of the segment (DCT2) expresses both NCC and ENaC (22). We assume that along the DCT2, the expression of NCC decreases linearly, whereas that of ENaC increases linearly.

The interstitial fluid solute concentrations are shown in Table 2 at the cortico-medullary boundary, at the outer-inner medullary boundary, and at the papillary tip. In general, we assume that interstitial concentrations vary linearly between the cortico-medullary junction and the inner-outer medullary boundary and between the inner-outer medullary boundary and the papillary tip. The interstitial fluid in the cortex is taken to be homogeneous, with one exception: following the approach of Weinstein (37, 39), we assume that there is a NH₃/NH₄⁺ concentration gradient in the cortex.

Table 2.

Interstitial solute concentrations

Solute	C-M	OM-IM	Tip
Na+	144	299	349
K+	4.90	10.0	20.0
Cl−	117	280	345
HCO3−	25.0	18.7	9.38
H2CO3	0.00441	0.00441	0.00441
CO2	1.50	1.50	1.50
HPO42−	3.00	2.79	2.17
H2PO4−	0.900	1.11	1.73
NH3	0.0123	0.0376	0.0629
NH4	0.828	3.86	8.89
HCO2−	1.00	1.00	1.00
H2CO2	0.000273	0.000362	0.000724
Glucose	5.00	8.33	8.50
Urea	8.00	60.0	200
pH	7.32	7.20	6.90
Osmolality	311	693	963

Values shown in mM except for pH, which is dimensionless, and for osmolality, which is shown in mosmol/kgH₂O. C-M, cortico-medullary boundary; OM-IM, outer-inner medullary boundary; Tip, papillary tip.

Boundary conditions.

Tubular fluid concentrations at the proximal tubule inlet are equal to those in the local cortical interstitium, except for the absence of protein. The superficial glomerulus is assumed to be located 0.2 mm from the surface of the cortex, whereas the juxtamedullary glomeruli are assumed to be located near the cortico-medullary boundary. Owing to the NH₃/NH₄⁺ concentration gradient assumed in the cortex (see above), the proximal tubule inflow NH₃/NH₄⁺ concentrations differ between superficial and juxtamedullary nephrons.

Proximal tubule inflow fluid pressure is taken to be 11.3 and 12.5 mmHg, respectively, for superficial and juxtamedullary nephrons. SNGFR for the superficial and juxtamedullary nephrons is taken to be ∼30 and 45 nl/min, respectively. [In the companion study (19), SNGFR is computed via the tubuloglomerular feedback.] Given these boundary conditions and baseline parameters, tubular fluid pressures of all model nephrons are equal at the entrance of the cortical collecting duct, where the nephrons merge, and collecting duct outflow pressure is 3.5 mmHg. When model parameters are varied in ways that affect water transport, tubular luminal radii are adjusted, via an iterative procedure, to yield the baseline collecting duct outflow pressure (3.5 mmHg). Specifically, if the predicted collecting duct outflow pressure is <3.5 mmHg, luminal radii are slightly increased for all tubules and tubular pressure is updated (Eq. 17) and vice versa.

MODEL RESULTS

Base-case results.

Using baseline parameters, we computed steady-state solutions to the model equations. Figure 2 shows delivery of key solutes (Na⁺, K⁺, Cl⁻, HCO₃⁻, NH₄⁺, H₂PO₄⁻, and urea) and fluid to the inlets of individual nephron segments. Values are given per kidney, separately for superficial and juxtamedullary nephrons. Results for the juxtamedullary nephron segments were computed as the sum of values for all five types of model juxtamedullary nephrons, ∑_i=1⁵ n_JM−iD_JM−i,k, where D_JM−i,k denotes the delivery of solute k associated with the i-th juxtamedullary nephron.

Recall that the model assumes a superficial-to-juxtamedullary nephron ratio of 2:1 (17). Thus, if tubular flows were all equal, the delivery values of superficial nephrons would be twice those of juxtamedullary nephrons. Together with the assumption that juxtamedullary nephron SNGFR is ∼50% higher than superficial nephron SNGFR, total inflow into superficial proximal tubules is 33.3% higher than total inflow into juxtamedullary nephrons.

The model predicts that the proximal tubules of the superficial and juxtamedullary nephrons reabsorb 68.7 and 75.2%, respectively, of the filtered Na⁺, primarily via NHE3 and Na⁺-K⁺-ATPase, and 66.7 and 74.1% of the filtered Cl⁻. The fractional reabsorption values are higher for the juxtamedullary proximal tubules owing to their higher SNGFR (thus higher solute loads) and the assumption of higher transport capacity compared with the superficial proximal tubule. The majority of the remaining Na⁺ and Cl⁻ is reabsorbed along the thick ascending limbs, resulting in urine excretion fractions of 1.4 and 1.2% for Na⁺ and Cl⁻, respectively. Similarly, the model predicts that 67.6 and 74.5% of the filtered K⁺ is reabsorbed along the superficial and juxtamedullary proximal tubules, respectively; the majority of the remaining K⁺ is also reabsorbed along the thick ascending limbs. However, the model distal convoluted tubules and connecting tubules are assumed to vigorously secrete K⁺. Consequently, 16.9% of the filtered load of K⁺ is excreted in urine. A substantial fraction of NH₄⁺ is reabsorbed along the thick ascending limbs by substituting for K⁺ in the Na⁺-K⁺-2Cl⁻ cotransporter.

The model predicts that 75.3 and 80.5% of the filtered volume is reabsorbed along the superficial and juxtamedullary proximal tubules, respectively. Water reabsorption in the proximal tubule also drives paracellular urea reabsorption in that segment. More water is reabsorbed downstream, albeit at a slower rate. The model represents a kidney in antidiuretic state, with an inner-medullary collecting duct that is highly urea and water permeable. As a result, 41.5% of the urea flow is reabsorbed along the collecting duct. The urea reabsorbate is implicitly reflected in the interstitial urea concentration gradient that the model assumes along the medullary axis (Table 2), which results in the secretion of urea into the loops of Henle. Tubular and interstitial fluid concentration of key solutes, pH, and osmolality are shown in Fig. 3, A-F. As in the case of urea, the interstitial concentration of key solutes is taken to increase along the medullary axis for a kidney in antidiuretic state. The model predicts a urine osmolality of 771 mosmol/kgH₂O, and urine [Na⁺], [K⁺], [Cl⁻], and [urea] of 164, 70, 117, and 249 mM, respectively; electroneutrality is maintained by other ions, such as phosphate and bicarbonate. Urine pH is predicted to be 5.9, and urine flow is 0.41 nl/min per nephron, or 14.8 μl/min per kidney, assuming 36,000 nephrons per kidney.

Fig. 3.

Profiles of tubular fluid solute concentrations (A–D), pH (E), osmolality (F), volume flow (G), and fluid pressure (H). Solid black lines, superficial nephron. Blue dashed line, longest juxtamedullary nephron. Grey dashed-dotted lines, interstitial fluid solute concentrations, pH, and osmolality. PT, proximal tubule; SDL, short or outer-medullary descending limb; LDL/LAL, thin descending/ascending limb; TAL, thick ascending limb; DCT, distal convoluted tubule; CNT, connecting duct; CD, collecting duct.

Tubular fluid flow and pressure are shown in Fig. 3, G and H. The steep drop in fluid pressure along the papillary collecting ducts is due to their coalescence: in the inner medulla, the number of collecting ducts decreases exponentially (Eq. 2). Taken in isolation, this implies that collecting duct tubular flow (per tubule) increases exponentially, resulting in a steep drop in fluid pressure.

Figure 4 shows the T_Na and Q_O2 predicted for different nephron segments. Results for the juxtamedullary nephron segments are computed as total values for the five types of model juxtamedullary nephrons, similar to the delivery values in Fig. 2. Segmental T_Na, both active and total, and Q_O2^total are the highest along the proximal tubule. T_Na^active and Q_O2^total, computed per kidney, are comparable between the superficial and juxtamedullary proximal tubules: there are twice as many superficial proximal tubules, but the juxtamedullary proximal tubules have higher SNGFR and transport areas per nephron. The superficial thick ascending limb yields a T_Na^active that is 80.3% of the corresponding proximal tubule. However, the thick ascending limbs transport much less Na⁺ paracellularly (compare Fig. 4, A and B). As a result, they have a significantly lower Na⁺ transport efficiency compared with the proximal tubule. The model predicts for the superficial (juxtamedullary) nephron T_Na^total/Q_O2^total ratios of 19.6 (17.6) for the proximal tubule, 10.5 (10.0) for the thick ascending limb, 6.5 (5.5) for the distal convoluted tubule and 3.5 (3.6) for the connecting tubule (see Fig. 4D). The higher transport efficiency of the superficial proximal tubules, relative to the juxtamedullary ones, can be attributed to the flow-dependent transport capacity of proximal tubules (5, 6). Specifically, fluid pressure at the proximal tubule inlet is assumed to be lower in superficial nephrons than in juxtamedullary nephrons (11.3 vs. 12.5 mmHg). The lower luminal flow along the superficial proximal tubule reduces transcellular transport via the torque-dependent scaling, resulting in a smaller contribution of transcellular T_Na to overall T_Na, compared with the juxtamedullary proximal tubules.

Fig. 4.

Baseline Na⁺ transport (T_Na) and O₂ consumption (Q_O2), given per kidney. A: segmental T_Na^active, taken positive for reabsorption. B: segmental T_Na^total. C: segmental Q_O2^total. D segmental T_Na^total/Q_O2^total ratios. See Figs. 2 and 3 for additional notations.

The model predicts a whole kidney T_Na^active of 122 μmol/min, with 88, 32, and 1.7 μmol/min (or, 72, 27, and 1%) associated with the cortex, outer medulla, and inner medulla, respectively. Whole kidney T_Na^total is predicted to be 178 μmol/min, with 131, 37, and 10 μmol/min (or, 74, 21, and 5%) associated with the cortex, outer medulla, and inner medulla, respectively. Note that the largest absolute difference between T_Na^active and T_Na^total occurs in the cortex, which can be attributed to the substantial paracellular transport along the proximal tubules.

Varying SNGFR.

We conducted simulations where SNGFR was varied by ∼10 and ∼20% of baseline values. We assumed that changes in SNGFR are due to changes in renal blood flow, as opposed to changes in filtration fraction which are linked to changes in peritubular oncotic pressure. Thus we assumed no change in proximal tubule transporter expression levels, except in response to changes in torque. The predicted Na⁺, K⁺, Cl⁻, and volume deliveries to individual segments under differing conditions are shown in Fig. 5; predicted solute transport and Q_O2 are shown in Figs. 6, 7, and 8. When SNGFR is reduced by 10 and 20%, proximal tubule active T_Na is predicted to be 53.3 and 48.3 μmol/min per kidney, respectively (see Fig. 6A). These T_Na^active values correspond to a 8 and 17% decrease, respectively, from base case (57.9 μmol/min). Overall proximal tubule T_Na^total is predicted to be 114.7 and 106.5 μmol/min, respectively. These values, which correspond to a 6 and 12% decrease, respectively, from base case (121.6 μmol/min), exhibit glomerulotubular balance. Vice versa, with a 10 and 20% increase in SNGFR, overall proximal tubule T_Na^total increases by 5 and 9%, respectively.

Fig. 5.

Impact of varying single-nephron glomerular filtration rate (SNGFR) on segmental delivery of Na⁺, K⁺, Cl⁻, and fluid. Solute delivery given per kidney. Notations are analogous to Fig. 2. Insets: reproductions of distal segment values.

Fig. 6.

Impact of varying SNGFR on predicted Na⁺ transport (T_Na^active and T_Na^total), O₂ consumption (Q_O2^total), and transport efficiency (T_Na^total/Q_O2^total).Values computed per kidney. Notations are analogous to Fig. 4. Inset: reproduction of distal segment values.

Fig. 7.

Impact of varying SNGFR on predicted segmental K⁺, Cl⁻, and H₂O transport. Transport computed per kidney. Notations are analogous to Fig. 4. Insets: reproductions of distal segment values.

Fig. 8.

Nephron function obtained for base case and for −20 and +20% changes in SNGFR. White segments, outer-medullary values; bottom colored segments, cortical values; blue segments, inner medulla (values not shown). Whole kidney values are indicated above the bars.

The model connecting tubule exhibits flow-dependent transport (Eq. 21). At higher SNGFR, higher connecting tubular flow increases its ENaC expression and Na⁺ reabsorption (see Fig. 6, A and B). The elevated T_Na leads to increased K⁺ secretion (see Fig. 7A) and K⁺ excretion (see Fig. 5B). At sufficiently low luminal flow, the model predicts that the reduction in ENaC expression and Na⁺ reabsorption eliminates the transepithelial electrochemical gradient favorable to K⁺ secretion, and net reabsorption of K⁺ is predicted along the connecting tubule (see Fig. 7A).

As previously noted, proximal tubule transcellular T_Na is assumed to respond proportionally to changes in luminal flow (5, 40). Thus, as SNGFR increases, proximal tubule T_Na^active (and thus Q_O2^active) increases at about the same rate as T_Na^total (compare Fig. 6, A and B). As a result, the model predicts that the proximal tubule's transport efficiency (T_Na^total/Q_O2^total) does not change significantly as SNGFR increases (Fig. 6D). In contrast, the transport efficiency of downstream segments, with the exception of the collecting duct, increases significantly: as SNGFR and Na⁺ delivery to the thick ascending limbs increase, luminal [Na⁺] decreases more slowly along those segments. For example, at the outflow of the superficial thick ascending limb, luminal [Na⁺] is predicted to be 100, 66, and 44 mM, when SNGFR is 20% above baseline, at baseline, and 20% below, respectively. These trends continue through the initial segment of the connecting tubule. When luminal [Na⁺] is higher, paracellular Na⁺ reabsorption increases (or paracellular Na⁺ secretion decreases) more quickly than transcellular T_Na, because the latter is less sensitive to increases in luminal [Na⁺]; hence, overall T_Na^total/Q_O2^total increases (see Fig. 6D).

Along the connecting tubule, Na⁺ is secreted paracellularly. That paracellular secretion decreases as SNGFR increases. As noted earlier, at higher tubular flow, ENaC expression along the connecting tubule is assumed to increase. These two competing processes result in higher T_Na^total/Q_O2^total as SNGFR increases. At higher SNGFR, Na⁺ delivery and concentration at the entrance of the collecting duct are higher but so is tubular Na⁺ reabsorption. These competing factors yield nonmonotonic changes in luminal [Na⁺] at the entrance of the cortical collecting duct, which in turn yield nonmonotonic changes in T_Na^total/Q_O2^total, as SNGFR is varied. Taken together, the net result on whole kidney Na⁺ transport efficiency is an increase in T_Na^total/Q_O2^totalfrom 15.6 to 17.0 (the baseline value being 16.4) as SNGFR is varied from −20% below baseline to +20% above (see Fig. 8). A summary of key results is depicted in Fig. 9.

Fig. 9.

Summary of model results. Arrows indicate directions of transepithelial Na⁺ and K⁺ fluxes, omitted for segments without significant net transport. Arrow size indicates flux magnitude scaled by baseline value, although not exactly to scale. In most cases, higher SNGFR yields larger fluxes, except for the K⁺ fluxes along the connecting tubules.

DISCUSSION

Modeling approach.

In the present study, we expanded our epithelium-based model of water and solute transport along superficial nephrons (21) to incorporate juxtamedullary nephrons of varying lengths. Our mathematical model is conceptually similar to that of Weinstein (38, 39) but differs from it in several respects, including the convergence location of cortical and superficial nephrons, computation of tubular hydrostatic pressure and SNGFR, urea permeabilities of the inner-medullary loop segments, and flow-dependent transport along the connecting tubule and cortical collecting duct. Nevertheless, the two models yield similar predictions regarding whole kidney transport: fractional water and Na⁺ excretion is predicted to be 1.1 and 1.3%, respectively, of filtered load in our model, and to be 1.1 and 0.7%, respectively, of filtered load in Weinstein's model (39).

Differences between juxtamedullary and superficial nephrons.

In the present model, fluid pressures have to match at the cortical collecting duct inlet (i.e., where the 6 types of nephrons converge), and the pressure at the collecting duct outlet is fixed at 3.5 mmHg. Also, proximal tubule inlet fluid pressure is assumed to be 11.3 and 12.5 mmHg for the superficial and juxtamedullary nephrons, respectively. With these conditions, SNGFR of juxtamedullary nephrons is predicted to be ∼50% higher than that of superficial nephrons (11, 13, 15, 27). As described above, proximal tubule transport is proportional to tubular flow rate; in addition, at a given flow rate, the transport capacity of juxtamedullary proximal tubules is taken to be ∼75% higher than that of superficial proximal tubules, based on perfused tubules studies in the rabbit (14). For both reasons, fractional reabsorption is higher in the juxtamedullary segments than in the superficial ones. Aside from these quantitative differences, the transport behavior of the two types of nephrons is predicted to be similar (Fig. 2). Concentration profiles in juxtamedullary nephrons closely track those in superficial nephrons (Fig. 3).

Our main objective was to investigate how local changes in Na⁺ reabsorption affect Q_O2 and the efficiency of T_Na, as reflected by the number of moles of sodium reabsorbed per mole of oxygen consumed (i.e., T_Na^total/Q_O2^total). As recapitulated in Fig. 4D, T_Na^total/Q_O2^total along a given segment differs very little between superficial and juxtamedullary nephrons, except in the proximal tubules and descending limbs. In the former segments, owing to differences in tubular flow rate combined with the flow dependence of transcellular transport, the ratio of active-to-passive Na⁺ transport is higher in juxtamedullary proximal tubules than in superficial proximal tubules; hence, T_Na^total/Q_O2^total is lower in the former. In descending limbs, since Na⁺ is secreted via the paracellular route while there is no active Na⁺ transport, T_Na^total/Q_O2^total is negative. The short descending limb of the superficial nephrons in the inner stripe of the outer medulla is assumed to consist of a water-impermeable segment that spans the terminal 60% of its length, whereas the outer-inner medullary segment of the long descending limb is assumed to be highly water permeable (24, 35); both segments have low Na⁺ permeability (Table 1). In the lower half of the inner stripe of the outer medulla, where interstitial [Na⁺] is assumed to be high, the transmembrane Na⁺ concentration gradient favors Na⁺ entry into the short descending limb but primarily water reabsorption from the long descending limb segments. The lower 60% of the inner-medullary segments of the long descending limbs are assumed to be highly Na⁺ permeable (Table 1). However, most of the long loops turn back near the outer-medullary boundary and thus have rather short inner-medullary segments, which limits Na⁺ entry. As a result, overall Na⁺ secretion is significantly higher in the short descending limb, resulting in a more negative T_Na^total/Q_O2^total ratio.

Flow-induced transepithelial transport.

Solute transport along segments of the nephrons has been observed to vary as a function of luminal flow (32, 31, 18, 43). Following the approach of Weinstein et al. (40), we model a compliant proximal tubule, of which the transport density is scaled by the torque exerted on the microvilli. Because less data are available for the distal segments, we adopt a simple model for the flow-dependent transport along the connecting tubule and cortical collecting duct by assuming that their apical Na⁺ permeability increases linearly with tubular flow. With these assumptions, the model predicted substantial increases in T_Na, both active and passive, along the proximal tubule, connecting tubule, and cortical collecting duct, and in K⁺ secretion along the connecting tubule, as SNGFR was raised (see Figs. 6 and 7).

The model assumes that epithelial transport along other segments is independent of flow, owing to the lack of adequate experimental data. Some of those segments, e.g., the thick ascending limb, may well exhibit flow-dependent transport. Mascula densa cells have been shown to detect variations not only in tubular fluid composition (1) but in tubular fluid flow as well (33). Thick ascending limb luminal flow is known to induce the production of NO and O₂⁻ (2), which exert opposite effects on Na⁺ reabsorption: whereas NO inhibits thick ascending limb active Na⁺ transport via cGMP (25), O₂⁻ stimulates it (26). Since NO appears to be produced in excess of O₂⁻, an acute increase in flow may reduce thick ascending limb Na⁺ transport. However, the chronic setting may have a different effect. NO and O₂⁻ react to form ONOO⁻, which inhibits NOS 3 expression in the thick ascending limb (29), which may lower NO production, increase Na⁺ reabsorption, and decrease NaCl excretion. Another contributing factor may be adenosine, which is generated and released at enhanced rates when transport work increases. Adenosine has been reported to activate local tubular adenosine receptor subtypes to enhance transport in the cortical proximal convoluted tubule, but reduces it in the more hypoxic medullary thick ascending limb, consistent with a proposed role in metabolic control of renal function (34). That observation, taken in isolation, may suggest that any flow-dependent transport along the thick ascending limb may be less prominent than along the proximal tubule.

Sodium transport efficiency in the cortex and medulla.

Under baseline conditions, the whole kidney T_Na^total/Q_O2^total is predicted to be 16.4, close to the lower bound of the reported range of 15–25 in the rat (4, 7, 42). This overall number masks significant differences between cortical and medullary segments. As shown in Fig. 8, the rate of active T_Na (and active Q_O2) is about three times as high in the cortex than in the medulla, whereas the rate of passive T_Na is more than ten times higher in the cortex (compare the difference between T_Na^total and T_Na^active in Fig. 8). This result means that T_Na efficiency is significantly greater in the cortex.

As SNGFR increases, the ratio of active-to-passive Na⁺ transport decreases in most downstream segments and T_Na^total/Q_O2^total is augmented (Fig. 6). Specifically, a 20% increase in SNGFR is predicted to increase T_Na^total/Q_O2^total in the S3 segment and the medullary thick ascending limb by 4.9 and 4.5%, respectively; active Q_O2 is predicted to increase by 14% in the cortex (from 5.8 to 6.7 μmol/min, given per kidney) but only by 3.7% in the medulla (from 2.17 to 2.25 μmol/min). While flow-induced transport is represented along both the S1-S2 and S3 segments, that dependence is assumed to be substantially weaker (by 50%) along the latter (see mathematical model). Furthermore, the absolute increase in flow, and thus any flow-induced increase in transcellular transport, is much smaller along the S3 segment, relative to most of the S1-S2 segment. Thus opposite changes in T_Na^total/Q_O2^total are predicted for the S1-S2 and S3 segments. Conversely, a 20% decrease in SNGFR is predicted to lower active Q_O2 by 23 and 4%, respectively, in the cortex and the medulla. These results suggest that the medulla, and in particular the S3 and medullary thick ascending, which are vulnerable to hypoxia, are relatively protected from flow-induced increases in Q_O2, which may be relevant in pathophysiological conditions such as acute kidney injury or chronic kidney diseases.

GRANTS

This research was supported by the Department of Veterans Affairs (to V. Vallon) and National Institute of Diabetes and Digestive and Kidney Diseases Grants R01-DK-089066 (to A. T. Layton), R01-DK-56248 (to V. Vallon), and R01-DK-106102 (to A. T. Layton and V. Vallon) and University of Alabama at Birmingham (UAB)/University of California at San Diego (UCSD) O'Brien Center for Acute Kidney Injury Research Grant P30-DK-079337 (to V. Vallon).

DISCLOSURES

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

AUTHOR CONTRIBUTIONS

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