Theoretical effects of UTB urea transporters in the renal medullary microcircula - 《American Journal of Physiology》美国生理学杂志干细胞之家

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发表于 2009-4-21 13:45 |显示全部帖子 |倒序浏览 |打印

作者：Wensheng Zhang and Aurélie Edwards作者单位：Department of Chemical and Biological Engineering, Tufts University,Medford, Massachusetts 02155
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                     9 {" X5 A! U0 ]1 R6 ^# W

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         4 W& K, R8 l& h5 L6 A% |

                  ! Y  G9 D! d8 }; [3 v+ t: C



      【摘要】
   A mathematical model of transport in the renal medullary microcirculation was used to investigate the role of the UTB urea transporter expressed indescending vasa recta (DVR) endothelia and red blood cell (RBC) membranes. Oursimulations suggest that UTB raises RBC and plasma and interstitial ureaconcentrations by facilitating radial diffusion of the solute and therefore serves to increase the contribution of urea to the corticomedullary osmolalitygradient, assuming no secondary effects on tubular transport. However, bylowering transmural urea concentration gradients, UTB reduces water effluxfrom DVR through aquaporin-1 (AQP1) water channels, thereby decreasing plasmasodium concentration. The net result of these competing effects on theosmolality gradient depends on the fraction of filtered urea that isreabsorbed by vasa recta. We also found that the contribution of UTB to watertransport across DVR and RBCs is negligible, even in the absence of AQP1. Ourmodel predicts that UTB plays a significant role, however, in reducing theshrinking and swelling of RBCs as blood flows along the medulla.
      【关键词】 kidney vasa recta aquaporin water channels mathematical model transport# Z$ ^$ h3 R" ?# g& h1 J9 H0 `. d
   THE MICROCIRCULATION OF THE renal medulla plays a fundamental role by supplying oxygen and nutrients to the medulla, removing water andsolutes deposited into the interstitium by reabsorption from the loops ofHenle and the collecting ducts, and preserving the corticomedullaryconcentration gradients that are essential for the formation of concentratedurine. These functions are made possible by the countercurrent arrangement ofdescending vasa recta (DVR) and ascending vasa recta (AVR) and byultrastructural differences between DVR and AVR walls.0 x+ {+ L* V) T7 I7 ~
: ?  d5 Y* h' k4 i$ {* n* P
In the last decade, two families of urea transporters have been identifiedin the renal medulla, UTA and UTB. UTA isoforms are present in the collectingduct and the descending limb of Henle's loop, whereas UTB is found in DVR anderythrocytes ( 17, 24, 32, 34 ). The erythrocyte ureatransporter is most likely a complex channel with modifier sites on theextracellular surface, rather than a carrier( 28 )." k2 A, A# J* d1 d' Q) ^% T
6 s: G' e( ^8 |9 Z8 Y+ x6 r: g
Experimental studies with UTB knockout mice suggest that the ureatransporter plays a significant role in the urinary concentrating mechanism.Urinary osmolality was measured to be 25% lower in UTB-deficient mice than inwild-type mice ( 35 ); UTB was found to contribute significantly to the capacity of the kidney to concentrateurine and even more greatly to its ability to concentrate urea itself( 35 ). UTB allows red bloodcells and vasa recta walls to rapidly exchange urea, thereby preventing itfrom being carried away from the medulla by the microcirculation and enhancingthe corticomedullary osmolality gradients( 17, 28, 29 ). Experimental observationssuggest that some of the urea delivered to the tip of the papilla by UTAs andcarried up by the blood through AVR is not recycled by DVR lacking UTB but isreturned to the general circulation( 35 ). In addition, the ureatransporter is thought to play an important role in preventing large volumechanges in red blood cells during their transit in the medullarymicrocirculation, a hypothesis first developed by Macey and Yousef( 12 ).- K+ \! ^8 o2 J* o* P, k2 u- @
* p( K5 a0 k8 f0 `5 a6 h: E
Yang and Verkman ( 36 )reported a significant osmotic water permeability in Xenopus laevis oocytes expressing UT3 (a UTB isoform), suggesting the existence of acontinuous aqueous channel through the UT3 protein for both water and ureatransport. Sidoux-Walter et al.( 29 ) suggested that atphysiological expression levels in erythrocytes, UTB does not transport water. Whether UTB significantly contributes to transmural fluxes of water in DVR invivo remains unknown.

Questions also remain regarding possible differences between outermedullary (OM) and inner medullary (IM) urea transporters. Although antibodiesto UTB label the continuous endothelium of rat DVR( 32, 34 ), in vivo DVR permeabilitymeasurements suggest that inner medullary DVR may lack functional ureatransporters. Whereas the permeability of OMDVR to urea( P u ) is five times higher than that to sodium( P Na ) and is inhibited by addition of thiourea, phloretin,and p -chloromercuribenzenesulfonate, in IMDVR P u and P Na are closely correlated as a straight line with aslope of 1 originating from the origin and are unaffected by thiourea orphloretin ( 17, 24 ). It is possible, albeit unlikely, that unstirred layers in the renal interstitium might have obscuredthe P u and P Na of IMDVR in vivo.

The objective of this study was to use a mathematical model of transport inthe medullary microcirculation to gain some insight into the specific functionof UTB in descending vasa recta walls and red blood cells (RBCs).% b! A7 r% v2 h2 |

MODEL AND NUMERICAL METHODS, f$ j* Y% ?  c1 A. [# V) m9 `
# B# g) F) Z$ O" l
Glossary

A int Cross-sectional area of medullary interstitium
/ C% n0 L' K) h/ U! O4 ?
AVR Ascending vasa recta2 W8 z6 [' g' `$ L6 P0 }

Concentration of solute i in plasma, red blood cells, andinterstitium, respectively
/ l% V$ R0 l7 C4 R# _
D Diameter of vasa recta

DVR Descending vasa recta6 E% |/ Z) g- z7 l

f Equilibrium distribution coefficient of urea between red blood cellsand plasma; B, i: E, m6 y  V/ U8 h/ K
8 k" s! r: H  B. g; J
f i Fraction of filtered load of water or solute that is reabsorbed into the micro-circulation; @0 b, K1 h' q* r4 n) }
! ^# ^$ o8 J) J
IM Inner medulla) k6 w8 B1 m1 E: e4 \! l' _
$ ^6 N, _8 e2 c8 z
Flux of component i through pathway k in compartment j. _" N3 ~- _5 I2 v+ q9 \
1 u# z5 ?1 H' Q: }5 O# M2 C
L Length of renal medulla
# x0 n2 \2 S$ x/ ]
Hydraulic conductivity of pathway k in compartment j

N No. of vasa recta

OM Outer medulla: K0 o7 }; c4 B5 b

P Hydraulic pressure difference8 B! C5 w% O8 k9 L; O

Pe Peclet number8 r" g3 U& D! U5 k0 h2 Y8 u

Permeability of pathway k in compartment j tocomponent i

q j, Q j Volume flow rate in compartment j in a single vas rectum and in all vasa recta, respectively; q8 I$ n6 M3 f' u8 E- x
- W% o% Y; d( q* R
RBC Red blood cell
$ B  U2 l% x7 M: b. p
Greek Symbols
& d) n. z/ h  {  A/ p- O5 o) l! K
Red blood cell-to-vessel surface area ratio

i Activity coefficient of solute i0 }! r9 K2 t4 ?4 g# U

i Oncotic pressure due to solute i
3 m: j8 l4 Q; }2 _' e
i Reflection coefficient to solute i

i Generation rate of component i per unit volume ofinterstitium

Subscripts and Superscripts
  a: i! [3 U4 {+ @6 s. k8 h- d- M6 q
A Ascending vasa recta7 n& |4 J8 a- S% w; W4 y7 ^
) B1 B  F$ O. m5 x  W2 ?9 u
AQP1 Aquaporin-1 water channels
% g% A* Y, R: \
B Blood

D Descending vasa recta
8 `8 l# t, U$ O& G
I Interstitium

LM Lipid membrane of red blood cells7 y9 \. }5 _1 x5 Z' `

Na Sodium chloride1 _. N- f& U! @4 \7 H" ?1 ^

p, para Paracellular pathway2 ]! q4 g1 X7 @) z* e; F' H
+ f5 O, M5 b2 L. {! O* b# j
pr Proteins* b, g' u4 n" q# [, @( q

P Plasma
+ A$ m' v( s9 O
R Red blood cell0 l' \) b& `: j* y
# K) r; F5 _+ p6 {
u Urea. \6 Z2 E; J! W, ^4 j' D! e
% e6 `5 T+ R3 n0 X
UTB UTB urea transporter6 [6 H7 U- ?$ ?% y

v Volume

General Description of Model# x7 P5 y0 W! W5 o2 F; t
* T. N7 I5 F! T3 A2 q& p, G
The renal medullary microcirculation is a countercurrent exchange systemwith blood flowing down along DVR from the corticomedullary junction to thepapillary tip and looping back to the cortical veins along AVR. Duringtransit, water and solutes in blood can diffuse radially between vasa rectaand the medullary interstitium. This countercurrent flow configuration helpsto maintain corticomedullary osmolality gradients, which are essential for theformation of concentrated urine.
5 h& M' S2 i& _! R, G
Our model, which has been described and applied earlier( 2, 4, 39, 40 ), consists of steady-stateconservation equations for water and solutes in vasa recta and theinterstitium, coupled with expressions for fluxes across vasa recta walls andRBC membranes through paracellular and transcellular pathways. We onlyconsider those vasa recta that are destined for the IM, i.e., those that liein the center of the vascular bundles and do not perfuse the capillary plexusin the OM.# S6 W4 [6 ]% Z" Q( Q. M* G1 c

DVR and AVR exchange water, sodium chloride, urea, and proteins via severalroutes. Vasa recta walls are perforated by nonselective paracellular pathways,across which fluid transport is driven by Starling forces (i.e., transmembranehydraulic and oncotic pressure differences). In addition, two transcellularpathways have been identified in DVR endothelium and RBC membranes: aquaporin-1 (AQP1) water channels, which allow for water movement to theexclusion of all solutes, and UTB urea transporters.

One of the functions of the medullary microcirculation is to carry awaywater and solutes (such as sodium chloride and urea) reabsorbed from the loopsof Henle and collecting ducts. In this model, reabsorption into theinterstitium is accounted for by interstitial generation rates that undergospatial variation ( 2 ). In theOM, we assume that exchanges occur only between vasa recta and theinterstitium because DVR and AVR form vascular bundles from which nephronloops are excluded, so that generation rates are taken to be zero.. n" D; T" C7 @" g% I/ ^

Conservation Equations0 \: ?# V$ t# a- }* w6 S* z4 J

If x is the axial coordinate along the corticomedullary axis, conservation of volume in plasma and RBCs can be expressed as

( 1 )
  p% J, D( j0 J% I
( 2 )
$ a# Q2 A" p7 r) P: I& X0 R
where Q P and Q R are the plasma and RBC flow rates,respectively, and and are the volume fluxes (perunit membrane area) across vasa recta walls and RBC membranes, respectively.The parameter represents the cell-to-vessel surface area ratio, N denotes the number of vasa recta, D their diameter, and  and - apply to AVR and DVR, respectively. The second term on theright-hand side of Eqs. 1 and 2 accounts for the fact thatat various depths in the medulla, DVR break up to form a capillary plexus,from which AVR are formed and ascend. Hence, part of the flow is directlyshunted from DVR to AVR at various levels.9 J2 o1 f; p3 S7 B0 Z

Because the RBC membrane is impermeable to sodium chloride, proteins, andhemoglobin, conservation of sodium chloride and proteins in plasma andhemoglobin and other nonurea solutes in RBCs, yields, respectively
2 d$ c9 O6 A( k2 V! C3 K1 o
( 3 ); y4 R3 R" t# u2 [3 {

( 4 )2 H. j' X+ m: L. w+ X8 c" c$ k# B

where (para) is the paracellularmolar flux of solute i (per unit membrane area) from plasma tointerstitium, and and are the plasma and RBCconcentration of solute i, respectively. Conservation of urea, whichtraverses vasa recta walls as well as RBC membranes, can be written as

( 5 )
: r  Z) l! x4 B) y* v+ {) R: {
( 6 ). o) j3 M) j% [! i- l

where and are the molar fluxes ofurea through vasa recta walls and RBC membranes, respectively, and f is the equilibrium distribution coefficient of urea between RBCs and plasma,taken to be 0.86 ( 1 ).

Order-of-magnitude analysis suggests that axial diffusion in theinterstitium is negligible relative to radial transport ( 39 ). If reabsorption from theloops of Henle and collecting ducts is accounted for by interstitialgeneration rates, conservation of volume, sodium chloride, urea, and proteinsin the interstitium can be written( 2 ) as) n3 [: U. d- w- r# T

( 7 )
7 @6 o$ j/ Q5 `; `2 X- a! m* y
( 8 )

where i denotes sodium chloride, urea, and proteins, and and are the net fluxes of water andsolute i through the vasa recta wall. A int is thecross-sectional area of the medullary interstitium, and v and i are the local generation rates of volume and solute i, respectively, per unit volume of interstitium. The generation ratefor proteins is zero, assuming that proteins are exclusively exchanged betweenvasa recta and interstitium( 39 ). Calculations related to v and i are described immediatelybelow.

Reabsorption of Water and Solutes from Tubular System+ l, P# A/ t, q+ ~

Water and sodium chloride are reabsorbed from the loops of Henle and thecollecting ducts into the interstitium; urea is reabsorbed mostly fromterminal IM collecting ducts through UTA urea transporters ( 28 ). The reabsorbed water andsolutes must then be removed by the medullary microcirculation to avoidaccumulation in the medulla. The fractions of filtered volume, sodium, and urea that are recovered by vasa recta from the IM interstitium are denoted byf v, f Na, and f u, respectively. Baselinevalues for f v, f Na, and f u are taken as 1, 1,and 40%, respectively ( 2 ). If is the systemicconcentration of solute i and GFR is the glomerular filtration rate,the filtered load of solute i is. Hence,the overall amount of water or solute recovered by vasa recta (VRR) may beexpressed ( 2 ) as, i/ z& P: ~; F: i0 r$ C
/ P( s0 V, z8 o
( 9 )4 Y5 L7 T+ ]$ Z7 F1 w' u
/ T" B! C+ c) G3 i0 Q% `! c% a
where x IM is the axial position along the IM normalized byits length, A IM ( x IM = 0) is themedullary crosssectional area at the OM-IM junction, and should be set equal to onefor water.

Koepsell et al. ( 7 ) observedthat sodium concentration increases exponentially along the corticomedullaryaxis. Because the fraction of osmolality due to urea increases from 2 to50% between the corticomedullary junction and the papillary tip ( 19 ), urea concentration mustincrease even more rapidly. To yield a profile consistent with thoseobservations, the baseline expressions for the interstitial area-weighedgeneration rates are taken ( 2 )as
4 ?6 _! Q$ p( E( V6 G
( 10 )+ T/ E2 w; `  c6 c4 p( Z4 v0 G. ^

( 11 )9 I- a1 ?8 j9 {# F' K9 ~
2 w( z0 s# K! E$ r4 E
( 12 )$ g; ^! w% A  v: R0 C2 u; p
6 I6 }5 n7 n* j' `% j$ @& X
where,and are constants obtained bysubstituting Eqs. 10-12 into Eq. 9. The baseline valueof GFR is taken as 784 µl/s, and the systemic concentrations of sodium andurea are 150 and 5 mmol/l, respectively( 3 ). Thus we estimate thatwater, sodium, and urea are reabsorbed into vasa recta at a rate of 1.31 x 10 - 4 ml/s, 1.96 x 10 - 5 mmol/s, and 2.61 x 10 - 5 mmol/s, respectively.

Flux Equations' h+ S) J5 b2 X# T

As described earlier, there are three different pathways for watertransport across DVR endothelia. Hence, in Eq.1 is the sum of threecontributions, the fluxes through paracellular pathways, AQP1 water channels,and UTB urea transporters, respectively$ i8 Y/ `$ Q" x! B, u

( 13 )

( 14 )  w3 T1 P/ o* r3 r

( 15 )

where,and represent the hydraulicconductivities of the paracellular pathway, AQP1, and UTB, respectively.Superscript D signifies that, because AQP1 and UTB are expressed in DVR butnot AVR endothelia, the corresponding hydraulic conductivities in AVR aretaken to be zero. P is the transmural hydraulic pressure difference, pr is the transmural oncotic pressure difference due toplasma proteins, and pr is the reflection coefficient of theparacellular pathway to proteins. The interstitial concentration and theactivity coefficient of solute i are denoted by and i, respectively, and u is thereflection coefficient of the UTB urea transporter to urea.
2 G5 |$ H! P8 p0 m. ?
Equations 13-15 indicate that the driving force for water movement differs for paracellular and transcellular pathways. Indeed, thereflection coefficient of paracellular pathways to small hydrophilic solutesis negligible (i.e., there is no sieving of urea and sodium chloride acrossthese pathways), whereas that of AQP1 water channels and UTB urea transporters 0. Transmural sodium and urea concentration differences contributesignificantly to water flux across AQP1 and UTB, given the large value of RT (19.3 mmHg/mM). Because small solutes are more concentrated in theinterstitium than in DVR plasma, there is water efflux from DVR across AQP1and UTB; across paracellular pathways, however, sodium and urea exert noeffect, and Starling forces favor volume reabsorption into vasa recta (essentially because proteins are more concentrated in plasma than ininterstitium). Previous simulations in which water fluxes across UTB wereneglected suggested that there is a net efflux of water across DVR wallsthroughout the medulla. We found that a net amount of 1.7 x 10 - 4 cm 3 /s is transported frominterstitium to lumen across paracellular pathways, but twice that amount(i.e., 3.4 x 10 - 4 cm 3 /s) iscarried from lumen to interstitium across AQP1 water channels, so that theoverall amount of water exiting DVR is 1.7 x 10 - 4 cm 3 /s( 39 ).8 g) I, Z6 E0 y2 z/ l/ ~5 T7 ]( H' z

The volume flux across the RBC membrane,, is the sum of three terms,corresponding to AQP1 water channels, the lipid membrane, and UTB ureatransporters, respectively0 h$ x  g. m  C. ~) t
/ p4 c# }4 x& x( i0 t0 B
( 16 )8 q1 i; o; j. ]5 {) L+ U/ e
5 E: M( P: f( Y" a' y
( 17 )

( 18 )
7 A% Z2 B6 x. J3 G
where,and are the hydraulicconductivities of AQP1, the lipid membrane, and UTB in RBC, respectively, and pr and Hb are the oncotic pressures due toplasma proteins and to hemoglobin in RBCs, respectively.

The paracellular flux of solute i ( i = sodium, protein,urea) across vasa recta walls can be written as* R6 c. G2 D' j& h; f/ @9 T

( 19 ), f; ^7 r' L9 h/ }2 c! h3 f
0 l0 c0 n5 F0 s
( 20 )" N0 j: |5 p8 Z: K  ?1 E
7 j2 i' N8 k1 b( f3 L7 n
where (para) is the permeability ofthe paracellular pathway to solute i, and the Peclet number, Pe, is ameasure of the importance of convection relative to diffusion./ K2 I# M9 m4 X0 O4 f- o

Because UTB serves as a common channel for water and urea, the UTB-mediatedtransmural and transmembrane fluxes of urea are given by, respectively! }+ E: J, x. }9 e' Y4 `' H

( 21 )- [. P& a& x0 M! i* Y5 Y
; D6 |* l# }1 r6 V0 m+ a" |
( 22 )5 \8 h- Q+ ~+ g/ Q

where in Eq. 21, if 0 (i.e., thevolume flux through UTB is directed from DVR to the interstitium), and otherwise. Similarly, in Eq. 22, if 0 (i.e., thevolume flux through UTB is directed from RBC to the lumen), and otherwise. (UTB) and (UTB) are the ureapermeability of UTB in DVR walls and RBC membranes, respectively. In Eq.5, the overall flux of urea through the DVR wall,, is the sum of two terms:the flux through the paracellular pathway (i.e., Eq. 19 ) and thatthrough the UTB urea transporter in DVR (i.e., Eq. 21 ). Similarly,the overall flux of urea across the RBC membrane,, is the sum of the ureaflux through the UTB urea transporter in RBC (i.e., Eq. 22 ) and that through the RBC lipid membrane. The latter can be expressed as. ~+ e8 W7 P" @  c

( 23 )3 `6 r' p5 T' |; d9 Z' Y$ _) p* _
8 P, e2 P" |5 v, z
where (LM) is the ureapermeability of the RBC lipid membrane.; C5 ~9 ]* K4 K" e% w

Expressions for the cell-to-wall surface area ratio, the number of vasarecta, the cross-sectional area of the interstitium, and the relationshipbetween protein concentration and oncotic pressure are summarized in the APPENDIX. Transport and morphological parameters, as well asinitial values, are given in Table1. Parameters specifically related to the transport of urea andwater across UTB, also shown in Table1, are further described below.( t% Y6 N; ~7 k1 S$ {

Table 1. Model parameter values$ w& O1 D  K6 N" l7 z1 F- f* ?
0 q$ G4 u3 Q3 `  M9 N1 t
UTB Transport Parameters6 t: u" \1 d; |+ M

UTB permeability to urea. Yang et al.( 35 ) reported that, at 10°C, the urea permeability of the RBC membrane was 45-fold lower in UTBknockout mice than in wild-type mice. In the absence of data at highertemperature, we assumed that the urea permeability of UTB transporters, (UTB), and that of the lipidmembrane, (LM), are equal to44/45 and 1/45 of the overall permeability of RBC to urea, respectively, takenas 160 x 10 - 5 cm/s( 1 ). Hence, (UTB) = 156 x 10 - 5 cm/s and (LM) = 3 x 10 - 5 cm/s in our simulations. The former estimateis close to the measured urea permeability of the human RBC urea transporterhUT-B1 (i.e., HUT11), 1.2 x 10 - 3 cm/s( 14 ). In DVR, the ureapermeability of UTB transporters, (UTB), was taken as 285 x 10 - 5 cm/s, as reported by Pallone et al.( 24 ).
1 F; r# Y+ U2 F9 f+ |- O- H
UTB permeability to water. Yang and Verkman( 37 ) estimated that the waterpermeability of UTB transporters in RBCs, (UTB), is 0.145 x 10 - 2 cm/s at 10°C; because their resultssuggest a weak dependence of that permeability on temperature, we assumed thesame value at 37°C.- J9 J- Z& _( R

Direct measurements of the water permeability of UTB transporters in DVRwalls, (UTB), have not beenreported until now. Yang and Verkman( 37 ) found that, in RBCs, the single-channel permeability of UTB to water is similar to that of AQP1 waterchannels. We assumed that the single-channel permeability of UTB to water, aswell as that to urea, is identical in RBC membranes and DVR endothelia. Inaddition, we assumed that the overall water and urea permeabilities in RBC andDVR are the products of their single-channel value and the channel density,respectively. Comparing the overall permeabilities of DVR and RBC to water andurea, we obtain
: @/ I$ M( q6 f+ s1 q% S2 S, j7 _
( 24 )

where, as described above, (UTB) and (UTB) are the permeabilitiesof UTB in DVR endothelia to water and urea, respectively, and (UTB) and (UTB) are the permeabilitiesof UTB in RBC membranes to water and urea, respectively. With the use of thisapproach, the overall water permeability of UTB in DVR walls was estimated as2.64 x 10 - 3 cm/s.3 w+ P; d" u# m% z
2 ?* I* w  }; V. [! Y6 H
The water permeability and hydraulic conductivity of UTB in DVR endothelia[ (UTB) and, respectively] can berelated ( 36 ) knowing that

( 25 )6 H$ e$ P; R. B* Z
5 N( G+ t3 g% [) c9 Q6 l; x# B% a
where A is the surface area across which transport occurs, Cis the transmural solute concentration difference, and u w the partial molar volume of water, taken as 18 cm 3 /mol. Hence, is given by
$ p2 \7 t9 a2 G8 B& ~( u
( 26 )+ `' b0 c1 ^5 U( E, s
* t5 H# B6 E, u& b2 g& O$ V3 E
yielding.Similarly, was calculatedas 1.35 x 10 - 9 cm·s - 1 · mmHg - 1.+ u5 }+ c: ~* L# w
. F) I$ ^# B8 n, ]  M3 F. n
RBC lipid membrane permeability. Yang and Verkman( 37 ) estimated that AQP1 waterchannels and the lipid membrane account for 90 and 2% of all the water thatpasses through the RBC membrane, respectively, at 10°C; at 37°C, theyaccount for 79 and 15% of the water exchanged, respectively. Hence, we assumed that the lipid membrane-to-AQP1 water channel hydraulic conductivity ratio,,is equal to 15/79 at 37°C. Because the combined hydraulic permeability ofAQP1 and the lipid membrane has been reported as 22.8 x 10 - 3 cm/s at 37°C( 13 ), and it may be convertedto the hydraulic conductivity as 2.1 x 10 - 8 cm · s - 1 · mmHg - 1 from Eq. 26, and were estimated as 0.34 x 10 - 8 and 1.8 x 10 - 8 cm · s - 1 · mmHg - 1, respectively, at 37°C. Thevalidity of these assumptions was confirmed by using the same approach at10°C. At that temperature, if the ratio is taken as 2/90, and are calculated as 0.46 x 10 - 9 and 2.05 x 10 - 8 cm · s - 1 · mmHg - 1, respectively. The former value isin excellent agreement with the measured water permeability of RBC in AQP1/UTB null erythrocytes at 10°C, reported as 4.5 x 10 - 4 cm/s( 37 ), that is, 0.42 x 10 - 9 cm · s - 1 · mmHg - 1 in units of hydraulicconductivity.
9 ^" m$ H9 `( m5 Y# C) o4 k
Numerical Methods5 D- ~$ U- d; w7 C3 P( t0 n

Ordinary differential equations (ODEs) described by Eqs. 1-6 were solved along DVR and AVR to obtain the plasma and RBC volume flow rates(Q P and Q R ) and solute concentrations( and ), with initial values(i.e., in DVR at the corticomedullary junction) for all variables as specifiedin Table 1. Solving Eqs.1-6 requires determination of the fluxes (as expressed in Eqs.13-19 ), which are themselves a function of solute concentrations inDVR, AVR, and the interstitium. Hence, the conservation equations cannot besimply solved along DVR first and AVR afterward. Thus we first assumed valuesfor all variables throughout AVR, integrated Eqs. 1-6 along DVRand then looped back along AVR, where "guess" values were replacedwith new calculated values as integration proceeded. This integration processwas iterated until the values for all variables along AVR converged. At every step, hydraulic pressure and solute concentrations in the interstitium wereobtained by solving interstitial conservation equations ( Eqs. 7 and 8 ).' T+ _) R7 i3 L: H( x, I

The ordinary differential equations were integrated along vasa recta usingGear's method, which is a self-adaptive, multistep, predictor-corrector methodfor stiff ODEs. At each value of x, the system of four nonlinearalgebraic equations ( Eqs. 7 and 8 ) was solved using amodified Powell hybrid method, as described more fully in our previous work( 38 ).
% {) `5 r3 Z* N% h
RESULTS! [4 p2 N7 c5 t" a

In this model of the medullary microcirculation, the reabsorption of water,sodium chloride, and urea from the loops of Henle and the collecting ducts isaccounted for by interstitial generation rates, which vary in magnitude alongthe corticomedullary axis. The amount reabsorbed is expressed as the fractionof the filtered load that is removed by the medullary microcirculation,denoted by f v, f Na, and f u, for water,sodium, and urea, respectively. The baseline values forf v,f Na, and f u are taken to be 1, 1, and 40%,respectively.; B0 G' q8 D( X7 j0 ^
7 ]8 }1 }% X# u
Changes in the expression of UTB urea transporters and AQP1 water channelsin vasa recta will likely affect the entire urinary concentrating mechanism,including the reabsorption of water and solutes into the interstitium.Nevertheless, given the absence of specific experimental data and theconstraint that our model does not explicitly take into consideration tubular transport, the amount of filtered load that is reabsorbed into vasa recta wasassumed to remain the same with and without UTB in the simulations describedbelow.

In our baseline case, both AQP1 water channels and UTB urea transportersare expressed in DVR walls and RBC membranes. Illustrated in Fig. 1 a are thebaseline urea concentrations in RBCs, plasma, and interstitium near thepapillary tip. As blood flows along DVR, it encounters regions of increasing osmolality. Urea thus diffuses from the interstitium into DVR (there is also asmaller contribution from convection, as the model predicts that water isdriven into DVR across paracellular pathways by transmural oncotic gradients)( 38, 39 ), and from DVR into RBCs,raising concentrations in both compartments, albeit with a lag becausepermeabilities are not infinite. As blood ascends back to the corticomedullaryjunction, interstitial urea concentrations decrease, so that urea diffuses inthe opposite direction, from RBCs to AVR lumen, then to interstitium.
' K' r6 e4 u& g! I4 Y( \
Fig. 1. Variations in urea concentration in descending vasa recta (DVR) andascending vasa recta (AVR) plasma, DVR and AVR red blood cells (RBCs), andinterstitium along the corticomedullary axis toward the papillary tip. a, Baseline case; b, urea transporter (UTB) deleted from RBCmembrane; x, position along the corticomedullary axis; L,total length of the medulla.; J$ P/ i" n$ j
" O" t% {3 d4 g; a& D
Maximal Urea Transport Capacities" G: t; T8 l3 n0 K) ^

The maximum permeability of the RBC membrane to urea, in the limit of zerosolute concentration, has been reported as 1.6 x 10 - 3 cm/s( 14 ). In addition,measurements of the maximum urea flux through UTB in the RBC membrane rangefrom 0.8 to 2.5 x 10 - 7 mol ·cm - 2 · s - 1, asreviewed by Sands et al. ( 28 ).Using Eq. 24, we estimated that the maximum urea flux through UTB inthe DVR wall is between 1.4 and 4.4 x 10 - 7 mol · cm - 2 · s - 1. In our baseline results, the urea fluxesthrough UTB in RBC membranes and DVR walls are x 10 - 8 and x 10 - 8 mol · cm - 2 · s - 1, respectively, that is, at least 10times lower than saturated fluxes. Hence, urea transport through UTB does not appear to reach its maximal capacities under the conditions we considered.! t/ {2 E# E- U

Asymmetry of Urea Transport by UTB Channels
8 w/ ?! f0 m! V6 n* @/ h( j3 Z* v
Several investigators have reported that the net efflux of urea acrosserythrocytes at a given concentration difference is greater than the netinflux for an equal, but oppositely directed, gradient( 14, 28 ). While there could beasymmetry in the RBC membrane, transport asymmetry across UTB in the DVR wall is unlikely, because urea moves through both the apical and abluminalmembranes of endothelial cells, and the two sides are mirror images of eachother with respect to the urea diffusion pathway. In vitro measurements of thepermeability of DVR to urea based on diffusion from bath to lumen and lumen tobath were indeed found to be similar( 17 ). We therefore simulated transport asymmetry only across the RBC membrane. In the absence of specificmeasurements of UTB permeability to urea according to the direction oftransport, we investigated the effect of higher urea efflux permeabilities byeither increasing (UTB)10-fold during urea efflux [with (UTB) remaining equal to itsbaseline value during urea influx] or decreasing (UTB) 10-fold during ureainflux [with (UTB) equal toits baseline value during urea efflux]. Although there was a slight decreasein the contribution of urea to osmolality at the papillary tip in the lattercase (from 52 to 49%), the effects of such variations in (UTB) were otherwisenegligible.

If the direction of asymmetry is reversed, that is, if influx (UTB) is increased 10-fold,or if efflux (UTB) isdecreased 10-fold, the effects on medullary transport remain negligible.Because there seems to be no significant effect of UTB transport asymmetry ontransport across vasa recta and erythrocytes, and given the lack of specific experimental data, we did not take this asymmetry into account in theremainder of our simulations.4 A$ v; Z+ i6 |" H4 C5 N

Function of UTB as an Urea Transporter

To examine the effect of UTB on urea transport in the renal medullarymicrocirculation, we first investigated the effects on osmolality ofeliminating the urea transporter from DVR endothelia, RBC membranes, and both.As stated above, we assumed that the interstitial generation rates of water,sodium, and urea remained unaffected by deletion of UTB or AQP1. Shown in Table 2 are plasma osmolalityand u% (that is, the fraction of osmolality attributable to urea) at thepapillary tip, with and without UTB. Our results suggest overall that bymaking transport barriers more permeable, urea transporters significantly increase urea concentrations throughout the medulla. However, the UTB-mediateddecrease in transmembrane urea concentration differences is also predicted toreduce water efflux from AQP1 water channels, thereby increasing blood flow,decreasing plasma sodium concentrations, and possibly reducing osmolality.3 z5 ~7 v, n" A3 }$ X

Table 2. Effect of UTB urea transporters on papillary tip osmolality

Eliminating UTB from DVR walls. Our simulations indicate that ifurea transporters are eliminated from DVR walls, urea can only enter DVRthrough the more restrictive (i.e., less urea-permeable) paracellular pathway,urea influx is significantly reduced, and plasma urea concentrations decrease.Because the interstitial-to-plasma urea concentration gradient issignificantly greater, water efflux from DVR to the interstitium through AQP1water channels increases, thereby reducing blood flow and raising plasmaconcentrations. If reabsorption from the loops of Henle and the collecting duct remains unaffected, the net result is that the plasma concentration ofsodium chloride increases whereas that of urea decreases, so that thecontribution of urea to osmolality at the papillary tip is significantlyreduced (from 51 to 38% if f u is 40%).: m6 H4 t3 ^- Z' ]; S1 W

The variation in papillary tip osmolality itself depends on which of thesetwo competing effects dominates, the reduction in urea influx into DVR or theincrease in water efflux, as illustrated in Table 2. If f u remains equal to 40 or 60%, osmolality increases with the removal of UTB inDVR walls, because urea concentrations are high overall and the increase intransmural concentration gradients has a significant effect on water effluxand thus on plasma sodium concentration. If f u is taken as 20% bothbefore and after deletion of the transporter, the reduction in urea influxinto DVR more than compensates for the increase in vasa recta sodium chlorideconcentration, and osmolality at the papillary tip slightly decreases.; q' X2 e0 Y7 W5 G; S

Eliminating UTB from RBC membranes. Our model indicates that ifUTB is selectively removed from erythrocytes, RBC urea concentrations aresignificantly reduced given the lower membrane permeability. Due todiffusional (radial) equilibration between all compartments, plasma andinterstitial concentrations decrease as well, as illustrated in Fig. 1 b. However,there is a competing effect. Because DVR-to-RBC urea concentration gradientsincrease significantly, there is initially more water efflux from RBCs to DVR( Fig. 2 A ), which inturn translates into a higher efflux from DVR to interstitium to preserve thewater balance. The net result is a more significant volume efflux from DVR, asshown in Fig. 2 B,which raises plasma concentrations. Overall, the increase in the plasmaconcentration of sodium chloride is greater than (or equal to, iff u = 20%) the decrease in that of urea, so that osmolality at thepapillary tip increases (or remains unchanged, if f u = 20%,),whereas the contribution of urea is significantly reduced, as shown in Table 2. With f u =40%, papillary tip osmolality rises slightly from 1,077 to 1,106 mosmol/kgH 2 O when UTB is removed from RBC membranes, whereas u辌reases from 51 to 42%.

Fig. 2. Water fluxes with and without UTB in RBC membrane. The junction between theouter (OM) and the inner medulla (IM) corresponds to x / L =0.24. A : across RBC membrane in DVR. Water efflux through UTB isnegligible compared with that through non-UTB pathways [i.e., aquaporin-1(AQP1) and lipid membrane]. In the absence of UTB in RBC membrane, waterefflux through non-UTB pathways is much higher near the corticomedullaryjunction. B : across AQP1 in DVR wall. In the absence of UTB in RBCmembrane, the larger water efflux from RBC to DVR also increases water effluxfrom DVR to the interstitium, thereby concentrating plasma.

Even though the value of (LM) remains uncertain,sensitivity analysis suggests that this parameter has a very small effect onresults. (LM) was measuredas 1/45 of the overall urea permeability of the RBC membrane at 10°C( 35 ), but there are noreported measurements at 37°C. Because the permeability of the RBC lipidmembrane to water increases about sevenfold from 10 to 37°C( 35 ), it is possible that thepermeability to urea increases by a similar factor. We found that in thepresence of UTB, multiplying the baseline lipid membrane-to-overall ureapermeability ratio by 10 does not affect papillary tip osmolality or thefraction due to urea. If UTB is eliminated from RBCs, variations in (LM) have a small effect onosmolality. As described above, removing UTB in erythrocytes raises the plasmaconcentration of sodium chloride and lowers that of urea. However, the largerthe (LM), the smaller theseeffects because the contribution of the RBC lipid membrane to urea transportis comparatively larger. Hence, if the lipid membrane-to-overall ureapermeability ratio is 10 times the baseline value, osmolality at the papillarytip is 1,082 mosmol/kgH 2 O (with u% = 50%) without UTB in RBCs, and 1,077 mosmol/kgH 2 O (with u% = 51%) with UTB.

Eliminating UTB from both RBC membranes and DVR walls. Not surprisingly, when UTB is removed from both DVR walls and RBC membranes, thecombined effects are predicted to lead to a further decrease in u% at thepapillary tip. Whether osmolality increases or decreases depends on the ureareabsorption ratio, as described above. With our baseline value (f u = 40% before and after deletion), eliminating UTB urea transporters leads to a4% increase in papillary tip osmolality, whereas the fraction that is due tourea decreases more significantly, from 51 to 33%.6 q" n! m0 H) Z- @$ p' Q- ?

Our results suggest that UTB transporters greatly increase the diffusiveexchange of urea across vasa recta and erythrocytes and therefore raisemedullary urea concentrations in RBC, plasma, and interstitium, assuming thatreabsorption from the loops of Henle and the collecting ducts remainsunaffected; because the high permeability that UTB confers to membranes lowers transmural urea concentration gradients, the transporter also reduces waterefflux from DVR through AQP1 water channels, thereby lowering medullary sodiumconcentrations. If f u is at least equal to 40%, the increase in imparted by UTB ispredicted to be smaller than the decrease in, so thatosmolality at the papillary tip decreases overall.
1 I, Q  G1 Z/ I; F& b  _
Function of UTB Without AQP1 Water Channels

As our previous simulations indicated, AQP1 water channels in DVR favor theshunting of water from DVR to AVR in the OM, therefore reducing blood flowrate to the deep medulla and increasing osmolality( 18 ). More specifically,without AQP1 in DVR walls there appears to be no water efflux from DVR into the interstitium (except near the corticomedullary junction) because thebalance of forces (i.e., oncotic vs. hydraulic pressure differences) favorswater influx across the paracellular pathway throughout most of the medulla( 38 ). Plasma concentrations thus remain lower. Eliminating AQP1 water channels from RBC membranes alsodecreases osmolality, because reducing water efflux from RBCs to DVR lowerssolute concentrations in RBCs, and therefore in plasma and interstitium aswell. Indeed, given the countercurrent arrangement of vasa recta and thediffusive transport of urea, the concentration of urea in the interstitium (almost) always lies between that in AVR plasma and that in DVR plasma,whereas plasma urea concentrations are themselves bounded by RBC ureaconcentrations (see Fig.1 ).2 D0 S/ h* l8 c" z1 ?
1 K3 H* p: z0 ^
Yang and Verkman ( 37 )generated mice lacking both UTB and AQP1 and reported that the single-channelwater permeability of the urea transporter is similar to that of AQP1,suggesting that UTB could play a role in facilitated water transport. Our simulations also make it possible to selectively eliminate AQP1 water channelsand/or UTB urea transporters from DVR walls, RBC membranes, or both. Resultsare summarized in Table 3. Inthe absence of AQP1 water channels, small-solute concentration differenceshave no direct effect on volume fluxes in this model, so that there is noUTB-mediated reduction in water efflux, and therefore no decrease in sodiumconcentration. On the assumption that interstitial generation rates remain unaffected, our model predicts that without AQP1, UTB transporters increasemedullary urea concentrations in vasa recta and interstitium by enhancing thediffusive (radial) transfer of urea. As a result, both the osmolality at thepapillary tip and the fraction due to urea increase. In the complete absenceof water channels, the expression of UTB results in a 35% increase inpapillary tip osmolality, and u% increases from 50%, assuming that f u = 40% ( Table3 ).0 d7 |) [2 R: M8 g4 Y

Table 3. Combined effect of UTB urea transporters and AQP1 water channels onosmolality1 \) N' o3 I4 o( s7 A6 |( _
( m) O; g8 L2 `
Function of UTB as a Water Channel; E4 Y6 y9 L/ o/ Y' \

Verkman and colleagues ( 36, 37 ) have shown that UTB ureatransporters also function as water channels. Whether the contribution of UTBto transmural water fluxes is significant under physiological conditions hasbeen a matter of debate among investigators ( 29, 36, 37 ). Our simulations indicatethat the water flux across UTB in DVR walls and in RBC membranes is negligible relative not only to the transcellular flux across AQP1 water channels butalso compared with the water flux across the paracellular pathway in DVR walls(Figs. 2 A and 3 ). This is not surprising given that the hydraulic conductivity of UTB in DVR endothelia is estimated as2.46 x 10 - 9 cm ·s - 1 · mmHg - 1,as described above, whereas that of the paracellular pathway has been measuredas 1.8 x 10 - 6 cm · s - 1 · mmHg - 1 ( 23 ). Across DVR walls, from the corticomedullary junction to the papillary tip, the total amount of watertransported through UTB is calculated to be only 2 and 3% of that transportedthrough AQP1 and the paracellular pathway, respectively. As a result,eliminating the water transport property of UTB urea transporters in DVR walls(i.e., setting to 0 in Eq. 15 ) has a negligible effect on papillary tip osmolality, even inthe absence of water channels.' F  O& l' u9 }0 K2 s
: U& j3 v% l5 \% I8 ?1 \4 K
Fig. 3. Single-vessel water fluxes across the DVR wall in the baseline case. Theflux through UTB urea transporters is much lower (in absolute value) than thatthrough AQP1 water channels and through paracellular pathways. A negative fluxmeans that water transport is directed from the interstitium toward thelumen.1 h# O5 N) q* }7 l4 m

Our model predicts that across the RBC membrane, the fraction of watertransported through UTB transporters, the lipid membrane, and AQP1 waterchannels is 6, 15, and 79%, respectively, in close agreement with the valuescalculated by Yang and Verkman ( 37 ). Thus eliminating thewater transport property of UTB in RBC membranes also has an insignificanteffect on water flux through the RBC membrane in the presence of AQP1 water channels. Without AQP1, our simulations suggest that nearly 30% of the watercarried across erythrocytes goes through UTB transporters and 70% through thelipid membrane, so that setting the water permeability of UTB to zerosignificantly decreases water fluxes across RBCs. However, the effect onpapillary tip osmolality is small, as the latter decreases by and thefraction due to urea increases slightly. Hence, our results indicate thateliminating water transport through UTB in erythrocytes has a negligibleeffect on osmolality, with or without AQP1 water channels.% e/ M- G2 t% ?2 j0 d. v
0 X. L6 w$ h7 N  ^. e
The simulations above were based on a reflection coefficient of UTB to urea( u ) equal to 0.3, as estimated by Yang and Verkman ( 36 ). A zero reflectioncoefficient would indicate that urea has no effect on water flux across UTB(see Eq. 15 ), whereas u = 1 would mean that UTB isimpermeable to urea. Because this parameter is uncertain, we varied it between0 and 0.6.
. @  Q7 Q2 W5 S4 ?& J, a1 Z; B1 D
If u increases from 0.3 to 0.6, the net amount of watertransported across UTB from RBCs into the lumen throughout DVR increases from5.2 x 10 - 6 to 5.5 x 10 - 6 cm 3 /s, and that transported fromthe lumen into the interstitium throughout DVR increases from 5.5 x 10 - 6 to 7.1 x 10 - 6 cm 3 /s; the latter figurerepresents only 2 and 4% of the net amount of water transported across AQP1and the paracellular pathway throughout DVR, respectively. It is notsurprising that changes in u have small effects because thehigh permeability to urea imparted by UTB results in small transmural osmoticpressure gradients due to urea; most of the driving force for water transport across UTB stems from other solute concentration differences. Thus if u is either increased twofold or set to zero, variations inthe amount of water transported through UTB in DVR walls have little effect:the osmolality at the papillary tip and u% remain the same (1,078mosmol/kgH 2 O and 51%, respectively, with f u = 40%).

We assumed in our baseline case that the reflection coefficient of UTB tononurea small solutes (i.e., sodium chloride and other RBC nonurea solutes) isequal to 1; that is, the transporter is impermeable to these other solutes. Itis possible, however, that UTB constitutes a shared pathway for water, urea,and other solutes. If the reflection coefficient of UTB to nonurea smallsolutes is taken as 0.3 (that is, equal to that to urea), our simulationsindicate that the net amount of water transported across UTB from lumen tointerstitium throughout DVR decreases significantly, from 5.5 x 10 - 6 to 2.5 x 10 - 6 cm 3 /s. The direction of watermovement through UTB is even reversed across the RBC membrane, becausesmall-solute concentration differences then play a lesser role and oncoticpressure differences constitute the main driving force; whereas the net amountof water transported across erythrocyte UTB throughout DVR is calculated to be5.2 x 10 - 6 cm 3 /s from RBC intothe lumen in the baseline case, it is 2.8 x 10 - 5 cm 3 /s from the lumen into RBC ifthe reflection coefficient of UTB to nonurea solutes is taken as 0.3. Theoverall amount of water transported across RBC membranes is predicted to dropby only 1% nevertheless, as nonurea solutes in RBCs exert a smaller osmoticpressure and more water is then carried out across AQP1 as a result; osmolality at the papillary tip and the contribution of urea are found toremain the same as in the baseline case. It should be noted that varying thereflection coefficient of the transporter to nonurea solutes has a greatereffect on water fluxes than varying that to urea because DVR walls and RBCmembranes are much less permeable to these other solutes, so thatcorresponding transmural concentration differences are significantly larger and more important as a driving force.

UTB and the RBC Osmotic Balance
# ^. g. j% {8 z6 n4 v/ d
Several investigators have proposed that erythrocyte UTB urea transportersplay an important role in balancing the osmotic pressure on each side of thebarrier (i.e., RBC and plasma), thereby limiting the extent to which RBCsshrink along DVR and swell along AVR( 11, 12 ).  w+ t( M7 i" t& n. \

To examine this assumption, we predicted relative RBC flow rate variationsalong the corticomedullary axis in a single DVR and AVR, with and without thepresence of UTB in RBC membranes. Our results suggest that UTB indeed reducesthe magnitude of RBC volume changes along vasa recta, as illustrated in Fig. 4. In the presence ofAQP1, as blood flows from the corticomedullary junction to the papillary tipalong DVR, RBC volume decreases to 63 and 55% of its initial value with andwithout UTB, respectively. It then increases back to 100 and 104% of itsinitial value, respectively, as blood returns to the cortex along AVR. The effects of UTB on RBC volume are slightly smaller in the absence of AQP1 inthe RBC membrane, because water efflux across erythrocytes is reduced withoutthe high osmotic water permeability imparted by AQP1 water channels. In thiscase, we found that along DVR, RBC volume decreases to 73 and 65% of itsinitial value with and without UTB, respectively; it then returns to 100 and102% of its initial value, respectively, as blood flows back to the cortex.
; y$ y. [  f: o" ~& F
Fig. 4. Single RBC volume along DVR and AVR divided by that at the corticomedullaryjunction in DVR (initial value). Presence of UTB in RBC membrane helps toreduce transmembrane osmotic pressure differences, so that RBC volumevariations are smaller.5 P5 f8 c8 ~: P$ d% R

Urea vs. Sodium Chloride DVR Permeability3 G/ d) O9 d, N+ \0 K- X

Pallone et al. ( 22 )suggested that a tradeoff may have evolved in the medullary microcirculation.The authors noted that whereas the urea gradient across the DVR wall isprobably small due to the high permeability imparted by expression of the UTB transporter, the permeability of at least some DVR to sodium chloride is low.Relatively low DVR sodium chloride permeability would favor the bypassing ofwater from DVR to AVR via AQP1, the purpose of which may be to lower bloodflow rate toward the papillary tip. A reduced blood flow rate to the deepest portions of the medulla is expected to enhance the efficiency of microvascularexchange in the IM by reducing solute washout.

Our model indicates that both sodium chloride and urea contribute significantly to water efflux across DVR. As illustrated in Fig. 5, interstitial-to-DVRconcentration gradients are higher for sodium chloride than for urea in partsof the IM but lower in the OM. Indeed, because the initial concentration ofurea is 50-fold lower than that of sodium, because the fraction of filteredurea that is reabsorbed by vasa recta is 40 vs. 1% for sodium in our baselinecase, and given that the interstitial area-weighted generation rate in the IMis assumed to increase exponentially for urea but only linearly for sodium( 2 ), the concentration of ureaincreases much faster than that of sodium in the OM and transmural gradientsare correspondingly higher. In the IM, the high DVR permeability to urea playsa more dominant role and reduces interstitial-to-DVR urea concentrationgradients.1 e- d" k. i9 X0 `
) B: Y) a8 h% y
Fig. 5. Interstitial-to-DVR NaCl and urea concentration differences with fractionof filtered urea (f u ) = 20, 40, and 60% (the arrow is pointedtoward increasing f u values). The concentration difference for NaClis 2-fold that for sodium.
/ ^/ M2 |2 N) Q# n. ]* B
Thus the contribution of sodium to water efflux across AQP1 water channelsappears to be more significant than that of urea in the IM, but the oppositeis true in the OM. Decreasing the urea reabsorption ratio to 20% reducestransmural urea concentration differences( Fig. 5 ), but the latter stillplay a larger role in driving water out of DVR in the OM.
2 `3 g" N& S" \, s- C
If the permeability of DVR to sodium were as high as that to urea, oursimulations indicate that water efflux from DVR to the interstitium throughAQP1 would be lowered, blood flow at the papillary tip would significantlyincrease as shown in Fig. 6,and plasma urea concentration would decrease. However, because a higherpermeability to sodium increases transmural (i.e., radial) transport of sodiumand thereby significantly raises sodium concentration, the osmolality at thepapillary tip would increase from 1,077 to 1,134 mosmol/kgH 2 O iff u remained equal to 40% (and from 756 to 830mosmol/kgH 2 O with f u = 20%). The contribution of urea topapillary tip osmolality would fall from 51 to 45% with f u equal to40% (and from 34 to 29% with f u equal to 20%). Although it mayfirst appear surprising that the presence of UTB (i.e., a large DVRpermeability to urea) slightly lowers osmolality at the papillary tip iff u is at least 40% whereas increasing the DVR permeability to sodium has the opposite effect, the results here again depend on the fractionof filtered solute that is reabsorbed by vasa recta. If the fractioncorresponding to sodium is decreased from 1 (i.e., our baseline value) to0.5%, increasing DVR permeability to sodium as described above would have aninsignificant effect on osmolality at the papillary tip. Our model shows thatincreasing the permeability of DVR to a given solute i gives rise totwo competing effects: higher plasma concentration of solute i due tomore efficient radial transport, and lower concentration of all other solutesdue to reduced transmural gradients and thus less water efflux from DVR. Whicheffect dominates depends on the amount of solute reabsorbed into themicrocirculation.
: ?& q: R2 w8 E, @% f8 Z3 W7 q
Fig. 6. Effect of sodium permeability ( P Na ) on single DVRplasma flow rate relative to that at the corticomedullary junction. In thecase other than the baseline, overall DVR P Na is taken tobe equal to that of urea. A higher P Na decreasestransmural NaCl concentration gradients, so that less water is shunted fromDVR to AVR, and plasma flow rate increases.

In summary, our results confirm the hypothesis of Pallone et al.( 22 ) that the low permeabilityto sodium chloride measured in some DVR may serve to enhance water transportfrom DVR to AVR across AQP1, thereby lowering blood flow to the papillary tip.Assuming that interstitial generation rates remained unchanged, we found thatif permeability of DVR to sodium were equal to that to urea, blood flow rateat the papillary tip would indeed increase slightly (by 10% compared withour baseline case). It is likely, however, that varying permeability of DVR to sodium affects reabsorption from the loops of Henle and the collecting ducts,so that the in vivo effects of changes in sodium permeability are difficult topredict.% F. W! |) u$ E1 t
0 c7 T0 i) }. q( q7 |# X
UTB Urea Transporters in OM vs. IM$ ^1 e! n6 `# x$ g- q1 {8 E4 o& a

As discussed by Pallone et al.( 22 ), in vivo DVR permeability measurements suggest that IMDVR may lack a functional urea transporter. Wethus examined the effect of selectively removing UTB from IMDVR or OMDVR,assuming here again that deletion of the transporter does not affectreasborption from the loops of Henle and the collecting ducts. As shown in Table 4, our model predicts that papillary tip osmolality is lowest when UTB is only present in OMDVR andhighest when the transporter is only present in IMDVR. These intriguingresults stem from the competing effects of the urea transporter on sodium andurea concentrations. As illustrated in Fig.7 A, in the OM, is calculated tobe the same whether UTB is present throughout DVR or in OMDVR only, and it isalso the same (but lower) whether UTB is present in IMDVR only or not at all.In the IM, however, the rate of increase in is reduced whenUTB is present in OMDVR only, because the permeability of vasa recta to ureais suddenly decreased. Conversely, the rate of increase in is augmented inthe IM when UTB is present in IMDVR only. As a consequence, urea concentration at the papillary tip is highest when UTB is present in IMDVR only and lowestwhen it is only found in OMDVR, with intermediate values if the ureatransporter is either present throughout the medulla or entirely absent fromDVR walls ( Fig. 7 A ).Plasma sodium concentration, meanwhile, increases steadily as the urea transporter is removed first from OMDVR and then from IMDVR as well( Fig. 7 B ), due to theprogressively larger water efflux from DVR. The net result is that, relativeto the baseline case, osmolality at the papillary tip decreases if UTB ispresent in OMDVR only and increases if UTB is present in IMDVR only ( Table 4 ).
. g# c' [& Z" t6 k
Table 4. Selective effects of UTB urea transporters in outer and inner medullaryDVR- L; {) }* A) P* q
/ C% j- ~+ Q( F( O5 V7 B
Fig. 7. Small-solute concentration [urea ( A ) and sodium ( B )] inDVR plasma, relative to that at the corticomedullary junction in DVR, in 4cases: with UTB urea transporters present throughout the medulla; in OMDVRonly; in IMDVR only; or without UTB.8 J  b- ~4 a- b8 s- e
& o; f2 s# ]8 c  Q1 _
Our simulations thus suggest that the effects of the urea transporter areas significant in the IM as in the OM. Assuming that f u remainsequal to 40%, selectively removing UTB from IMDVR reduces osmolality at thepapillary tip by 5.5% (from 1,077 to 1,018 mosmol/kgH 2 O), whereaseliminating UTB from OMDVR only increases it by 5.2% (from 1,077 to 1,133mosmol/kgH 2 O).4 ^2 F& _) v1 O

DISCUSSION
& z5 ^1 J, x7 s- S! O* H1 R& ^. v. I
The urea transporter UTB has been identified in the membrane oferythrocytes and in the endothelium of renal medullary DVR. The transporterhas been reported to play an important role in regulating urea concentrationin the kidney, and it appears to function as a common channel for both ureaand water ( 34, 35, 37 ).! i5 q# `7 f" T2 L( J3 U" v  u/ P
6 v( D6 b: w( u! }4 e
In this study, we used our mathematical model of the renal medullary microcirculation to gain more insight into the role of UTB. Our approach islimited in two important ways. First, to obviate the need to fully simulatethe urinary concentrating mechanism, the model specifies interstitialgeneration rates of water, sodium, and urea to account for reabsorption fromthe loops of Henle and the collecting ducts. Because it is not known how changes in the expression of UTB and AQP1 in DVR affect tubular transport, weassumed that the fraction of filtered load that is reabsorbed into themedullary microcirculation remains unaffected by deletion of the transporters.However, given that the entire countercurrent system of the medulla acts in anintegrated manner, changes in transmural fluxes are bound to have secondary effects on transtubular gradients in vivo and therefore alter supply to theinterstitium.
! ^! r% S# X- k- V/ N* f
It is likely that deleting UTB affects not only f u (and the spatial variations of the interstitial generation rate of urea) but also waterand sodium reabsorption, as they are closely linked. As reviewed by Sands( 26 ), hyperosmolality in the IM collecting duct can raise facilitated urea permeability and increase ureareabsorption; water diuresis also appears to enhance urea reabsorption; andurea is actively transported from the IM collecting duct to the interstitiumby "sodium-urea cotransporters" in rats on a low-protein diet. Inaddition, even the baseline value of f u is difficult to estimatebased on experimental data, as discussed previously( 2 ). Theoretical studies( 30, 31 ) suggest that f u is comprised between 20 and 60%, hence the range examined in this work and ourchoice of 40% for the baseline case.
  [. a+ x4 L9 \$ z
The predictive ability of our model is further restricted by the nature ofthe relevant experimental data. Whereas most of the morphological andtransport parameters used in this model come from measurements in rats,experiments in which the expression of UTB or AQP1 is deleted are oftenperformed in mice. Observations by Verkman and colleagues( 35 ) suggest that, in mice,UTB-dependent countercurrent exchange of urea in the renal medulla maycontribute to one-third of the total capacity of the kidney to concentrateurine and even more greatly to the ability of the kidney to concentrate ureaitself. In studies with UTB knockout mice, urine osmolality was 25% lower, plasma urea concentration was 30% higher, and urine urea concentration was 35%lower than in wild-type mice( 35 ). The medullary architecture of mice is different from that of rats, as mice do not haveshort-looped nephrons, but it has not been thoroughly described inquantitative terms in the literature. Thus defects in the concentratingability observed in UTB knockout mice cannot be directly compared with thepredictions of this model.
% y1 a/ Y" R, Z  h8 V. o2 {% e2 ?2 y! R
Despite these limitations, our approach can help in gaining anunderstanding of the function of UTB. AQP1 knockout mice have been shown tomanifest a severe urinary concentrating defect associated with defectivemedullary interstitial osmolality ( 10 ). Our model, withparameters derived from measurements in rats, accurately predicted thatdeletion of AQP1 leads to a substantial reduction of interstitial osmolality.Simulations suggested that DVR expression of AQP1 enhances medullary osmolar gradients by providing a route for volume efflux that shunts blood flow fromDVR to AVR, secondarily reducing blood flow to the IM( 18 ). The present study, whileimpaired by the lack of experimental data regarding the effects of UTBdeletion on f v, f Na, and f u in rats, may alsoprovide some insights into the role of UTB as a urea transporter.

Our simulations suggest that, by greatly facilitating transmembrane ureadiffusion, UTB significantly increases urea concentrations throughout themedulla. However, by decreasing radial urea concentration gradients, UTB alsoreduces volume efflux from DVR through AQP1 water channels and thereby lowersthe plasma concentration of other solutes such as sodium chloride. The presence of UTB therefore appears to increase the contribution of urea to thecorticomedullary gradient. Whether the UTB-mediated increase in theconcentration of urea compensates for the decrease in that of sodium chloridedepends on the fraction of filtered urea that is reabsorbed by vasa recta,f u; the net effect on overall concentrating ability (as measured byosmolality at the papillary tip) is not expected to be very significant. In the absence of AQP1, however, the reduction in transmembrane ureaconcentration differences imparted by UTB has no direct effect on waterfluxes, and our model predicts that the urea transporter significantlyincreases both papillary osmolality and the fraction of total osmolality thatis due to urea., s# g  B% \/ U' S3 j+ t+ t
9 {+ l( T2 V  z; M% e% _
AQP1 water channels, expressed by DVR endothelia( 16 ), have been shown to be atransport pathway across which small hydrophilic solutes such as sodiumchloride and urea drive water flux ( 20 ). UTB also appears tofunction as a water channel, with urea and water sharing a common aqueouspathway. Yang and Verkman ( 36 )first reported a significant osmotic water permeability in X. laevis oocytes expressing UT3 (a UTB isoform), suggesting the existence of acontinuous aqueous channel through the UT3 protein that passes both water andurea. Sidoux-Walter et al. ( 29 ) later found that atphysiological expression levels, the HUT11A transporter in humans (whose rathomologue is UT3) confers urea permeability to RBCs but not waterpermeability. They proposed that water transport activity in HUT11A-expressing oocytes occurs when the transporter takes another conformation at high densityin the oocyte membrane, allowing for water movement./ H  V. h9 U" m: Y5 ]" q4 p! A, F) a

In subsequent studies with knockout mice, Yang and Verkman ( 37 ) reported that thesingle-water channel permeability of UTB, 7.5 x 10 - 14 cm 3 /s, is similar to that ofAQP1. The authors found that, at 10°C, the erythrocyte osmotic waterpermeability was significantly reduced in AQP1-UTB-deficient mice comparedwith AQP1-deficient mice (0.045 x 10 - 2 vs.0.19 x 10 - 2 cm/s). There was, however, nosignificant difference at 35°C; at that temperature, 79% of water was transported through AQP1, 6% through UTB, and the rest through the lipidmembrane. The investigators also found that urine osmolality in doubleknockout mice was similar to that in AQP1 knockout mice. They concluded thatUTB functions as an efficient water transporter, but its absolute contributionto total water transport in normal erythrocytes is small because RBCs express many fewer UTB urea transporters than AQP1 water channels( 37 ).$ ]( q9 Q% |3 {$ X3 ^/ f
  D8 u0 x" @5 r
Our theoretical predictions regarding the function of UTB as a waterchannel confirm the experimental results of Sidoux-Walter et al.( 29 ) and Yang and Verkman( 37 ), namely, that theabsolute contribution of UTB transporters to water transport in normal erythrocytes is not significant. In our baseline case (i.e., both AQP1 and UTBare present in erythrocytes and vasa recta), the total amount of watertransported across DVR walls through UTB is calculated to be only 2 and 3% ofthat transported through AQP1 and the paracellular pathway, respectively.Along DVR, the fraction of water transported across RBCs through the UTB transporters, lipid membrane, and AQP1 water channels is predicted to be 6,15, and 79%, respectively, in close agreement with the values reported by Yangand Verkman. It is thus not surprising that eliminating water transport acrossUTB should have a negligible effect on small-solute concentrations andosmolality, even in the absence of water channels.+ I# l# n6 Y! n  p
$ L! y* ?2 c" [# q1 K
Yang and Verkman ( 36 )speculated that the UTB-mediated solvent drag of urea could provide a way forurea to exit from vasa recta to balance the osmotically driven water exit.However, our simulations suggest that the convective efflux of urea from DVRacross UTB represents only 1% of the diffusional influx across the sametransporter (i.e., 8.9 x 10 - 7 vs. 8.5 x 10 - 5 mmol/s). The net amount of urea thatenters DVR from the interstitium through UTB is 8.4 x 10 - 5 mmol/s, which is also more than twice theamount that enters the lumen across the paracellular pathway, 4.1 x 10 - 5 mmol/s. The ability of UTB to transportwater, therefore, does not appear to significantly affect urea fluxes acrossDVR walls. Water movement through UTB is unlikely to be physiologicallyimportant in the kidney., ^+ e" Y* Z4 t/ \! x  X

As discussed by Macey and Yousef( 12 ), shrinkage of RBCs to whereby themembrane becomes leaky to sodium; conversely, swollen erythrocytes are lessdeformable and more prone to destruction. Our results confirm that the UTBurea transporter plays a significant role in erythrocytes by reducing themagnitude of RBC shrinkage and swelling along the corticomedullary axis. Wefound that with UTB, RBC volume decreases to 63% of its initial value along DVR and increases back to 100% of its initial DVR value along AVR; without thetransporter, volume would be reduced to 55% of its initial value along DVR andwould slightly exceed its initial DVR value on leaving medullary AVR. Thosetrends agree with the predictions of Macey and Yousef. These investigators estimated that the RBC volume at the papillary tip is 65 and 55-60%of its initial DVR value with and without UTB, respectively, and that as bloodflows back up, RBC volume returns to 100 and 130-145% of its initialvalue, respectively. The quantitative differences between the two studies maystem in part from the fact that their estimate of the urea permeability of theRBC lipid membrane is one order of magnitude lower than ours; Macey and Yousefused reported values for art& @: ?8 M* O7 W
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