Math1 Contput. Modding. Prmted in Great Britain
Vol. 14, pp. 538-542,
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IN THE RENAL
Raymond Mejia and Mark A. Knepper Laboratory of Kidney and Electrolyte Metabolism, NHLBI and Mathematical Research Branch, NIDDK National Institutes of Health, Bethesda, MD 20892 Abstract. A mathematical model of acid-base transport in a renal tubule is described. Solutions for validation and control simulations are given, and the model is used to show that in the thick ascending limb the NH3 permeability must be relatively small - on the order of 3 x 10-s cm/s - in order for the rat kidney to concentrate NH:. Keywords Mathematical model; acid-base balance; reactive and nonreactive port; transmural electrical potential; thick ascending limb of Henle’s loop.
to that for sodium ion has been proposed, and active absorption against a concentration gradient has been demonstrated by Good et al. (1984) in the thick ascending limb (TAL). Our model permits study of the function of the cortical (CTAL) and medullary thick ascending limb (MTAL) in acid-base balance, and allows us to test various hypotheses, including the effect of endogenous carbonic anhydrase in the lumen in the presence and absence of H+ secretion.
INTRODUCTION In mammals, extracellular pH is tightly regulated at about 7.4, and the kidney is instrumental in maintaining this homeostasis. The roles of the liver and kidney in regulation of systemic pH are shown in Figure 1. Excess amino acids derived from dietary proteins and tissue catabo!ism are utilized in the liver for gluconeogenesis. Byproducts of this process, derived from the amino and carboxyl groups of the amino acids, are NH: and HCO;. Additional byproducts are sulfuric acid, which is derived from sulfur containing amino acids, and phosphoric acid, derived from phosphorylated amino acids. These are strong acids that constitute a major acid load to the extracellular fluid, titrating HCO, and other buffers. The kidney not only excretes NH:, HzPO; and SOibut is a source of HCO, that neutralizes H+ in the extracellular space. This requires direct NH,f transport in some nephron segments and a combination of NH, and H+ transport in others, as well as HCO, synthesis and excretion (Knepper et al., 1989).
MODEL The mathematical model is described by time dependent equations for solute and fluid conservation, equations for electroneutrality and buffer balance (see Star et al., 1987 and Mejia, 1988) and equations of motion in each tubule segment as follow
= -Jk + A& (solute conservation)
where z is axial distance; t is time; A is the crosssectional area of the segment; Ck is the concentration of the k”’ species; F, is volume flow; Dk is the diffusion coefficient of the /cthspecies; $J is the electrical potential; Jk is solute flux (positive defined to be out of the lumen), and Sk is the production or consumption of the klh species due to chemical reaction. Species mobility uh is related to the diffusivity by
Ammonium is accumulated in the renal medulla to concentrations far exceeding those in the systemic blood or the renal cortex. Countercurrent multiplication similar
where tk is the valence of the kth species; F is Faraday’s constant; R is the gas constant, and T is the absolute temperature. c?F” aA aZfat=-J” URINE
(fluid conservation) where J, is volume flux out of the lumen.
Fig. 1. Roles of liver and kidney in regulation of systemic pH (Knepper et al., 1989). * Sites of regulation; a - KG2- is ol-ketoglutarate.
=-=O (electroneutrality) 538
Proc. 7th Int. Conf on Mathematical and Computer Modelling where z is the vector of valences for all species; C is the vector of concentrations, and J is the vector of solute fluxes.
(Henderson-Hasselbalch) where pH = - logCH+; pKB = - logKe; the dissociation constant Kg is the proton concentration at half neutralization; subscripts B- and HB designate the base and protonated forms of buffer pair B, respectively.
-& < z,J >=0 and solved for the electrical
Z+&F,=o where P is hydrostatic to flow.
an IBM PC or clone, due to Kahaner and Barnett (1989). To use PLOD, the dissociation constant in each Henderson-Hasselbalch equation is replaced by a ratio of forward and backward reaction rates, and buffer balance is incorporated into the source term of the solute conservation equations. This leads to a stiff system of differential equations that is solved with the Gear op tion in PLOD using variable order integration up to order five. To complete the system of differential equations, electroneutrality is restated as
in the lumen.
We describe active secretion into the lumen (either from a bath or interstitium) as a function of the recig rocal of the luminal concentration. Thus active proton secretion becomes
and R, is resistance
water flux is given by with V,,,,+ negative.
where p is the radius of the tubule; L, is the hydraulic permeability. ok is the reflection coefficient of the kth species, and ACI, = CI, - C;, where Cl is the concentration of the kth species in the interstitium. The transmembrane given by
flux for neutral
where Pk is the solute permeability of the krh species, and & = (C, t C;)/2. Active transport is defined using Michaelis-Menten kinetics as follows vw.kck ”
ck - c;6-rrq 1 -
(1 - (rk)&,J” + J;
where q = FA$IRT, and A$ = G - v,!J’, where 4 and Q!J* are the electrical potential in the lumen and interstitium, respectively. Chemical
sources of carbonic
The parameters of the model are given in Table 1, with anatomic parameters taken from measurements in the rat. k-1 is given in the table for the case where carbonic anhydrase (CA) is absent from the lumen. When carbonic anhydrase is present, kkl is taken as 49 x lOs, which is sufficiently large to simulate the presence of the enzyme. Transport coefficients, taken from rat data wherever available, are given in Table 2. Reflection coefficients gk = 1 for all k. Diffusive flow in the lumen is negligible in comparison with bulk flow, so we set Dk = 0 for all k. The NH4f permeability is that reported in rat by Good (1988). In the cortex, Na+ and C1- permeabilities are those measured in rabbit by Burg and Green (1973); K+ permeability is that reported by Shareghi and Agus (1982). In the medulla, Nu+ and C’- permeabilities are those measured in rabbit by Rocha and Kokko (19’73); K+ permeability is that measured in rat by Work and
acid and CO* are given
by SHICOs = k,[COa] - k_,[H*COs]
where kr is the hydration rate constant of COZ, and k-, is the dehydration rate constant of HrCOa. METHOD
Kmk + Ck
where v,k is the maximum rate of active transport the kth species, and K,,,k is the Michaelis constant. the valence %k # 0, Jk = %,d&?kq
Initial conditions are prescribed for the lumen, and the variables in the interstitum are obtained analytically from boundary data using a linear profile. The transluminal hydrostatic pressure gradient has a second-order effect on the solution, so we neglect it altogether.
Table 1: Parameters P (pm) 13.5 CTAL length (cm) 0.2 MTAL length (cm) 0.2 PK HCO;/CO, 6.10 H&OsIHCO; 3.57 NH.? INH, 9.03 I Reaction Rates (s-l) Species 1 k-1 ] kr H2C0,/COr / 49. 1 144.2 I,
Steady-state solutions for these equations have been obtained using PLOD, a computer program for the so lution of systems of ordinary differential equations on
Proc. 7th Inr. Conf. on Mathematical and Computer
Schafer (1983). The maximum rate of active transport of nonreactive species is that used by Stephenson et al. (1987). The maximum rate of NH: transport is that measured by Good et al. (1984). Solutions are insensitive to small variations in the Michaelis constants, which are chosen to be consistent with the sodium and chloride outflow concentrations.
co2 NH: H+ HzCOs NH, Urea
cm 6.00D-5 0. 0. 3.10D-3 0.86D-5
10. -10. 0. 3;;.8.
/ Z,“,“D”:; j
j 50. 1
0.4 cm. The interstitium is replaced by a well-mixed bath as done by Star et al. (1987). The lumen is perfused slowly (Q = 1.35 nl/min at z = 0.4 cm) with a perfusate that is at nearly zero potential relative to the bath. A standing gradient is reached in each tubule segment with [Na+] at 50 mM and a potential difference of 12 mV at the cortical end. Of course, in a laboratory experiment, only values at the perfusion and collection ends are known, so the plateau at the junction of the cortical and medullary segments would not be observed. Figure 3 shows how we estimate the transport potential which results from active transport of Na+ and Cfwith a 1:2 stochiometry. Sodium and chloride concentrations are plotted versus the electrical potential difference. The active transport potential is the A$ at which transport occurs with no concentration gradient. The lumen has been perfused rapidly, so that the effect of active transport dominates. In this simulation the potential difference at the perfusion end is 5 mV, and the composition of nonreactive ions in the perfusate has been adjusted until the INa+] and [Cl-] in the bath and lumen are equal at the same A+, which is 10 mV. This is consistent with measurements by Knepper (unpublished data).
of the Model
Two validations that are commonly done in the laboratory are stop-flow and fast-flow experiments. The sodium and chloride concentrations in a simulation of a stop-flow experiment are shown in Figure 2. The cortical end of the thick ascending limb is at length 0; the junction of the cortical and medullary segments is at 0.2, and the junction of the thick and thin limbs is at
Fig. 3. Sodium and chloride concentration profiles versus electrical potential difference in a simulation of a fast-flow experiment. Potential difference increases monotonically from medulla to cortex.
Fig. 2. Sodium and chloride concentration profiles versus tubule length in a simulation of a stop-flow experiment. CTAL segment is between 0. and 0.2 cm, and MTAL is between 0.2 and 0.4 cm. TAM is total ammonia.
A control experiment is defined using the boundary and initial data shown in Table 3, and the equations that describe the lumen are solved against the interstitium. For nonreactive species the boundary values in the intersitium and the luminal composition at the junction of the inner and outer medulla are those obtained by Stephenson et al. (1987) in their simulations. Figure 4 shows the potential difference in the lumen. Figure 5 shows the concentration of sodium and chloride ions in the lumen, and Figure 6 shows the concentration of ammonium and total ammonia in the lumen.
Proc. 7th Int. Conf. on Mathematical and Computer Modelling Table 3: Boundary and Initial Data Interstitium Concentration (mM) Species Cortex Inner Medulla lz.0; 25. 25. NH3+NH,+ 0.1 1.2 CO2 1.2 1.2 Na+ 140. 283. K+ 4.5 9. Urea 5. 5. PB 7.42 7.42 V (mv) 0 -5. Thick Ascending Limb Concentration 25. 6. NH, + NH: CCZ 1.2 Na+ 280. K+ 10. Urea 10. PB 7.42 Cc,(mv) 0. Q (nl/min) -5. L, (cm/s/arm) 0.
Fig. 5. Sodium and chloride concentration profiles versus tubule length in the control simulation.
Fig. 4. Electrical potential difference versus tubule length is shown for a tubule immersed in an interstitium. This simulation is used as a control.
Fig. 6. Concentration of ammonium and total ammonia versus tubule leneth in the control simulation ~-
Figures 4,5 and 6 show that the electrical potential and the ion concentrations at the cortical end of the lumen are consistent with in vitro measurements for the rat and that most of the reabsorption of NH: occurs ill the medullary thick limb. An Experiment We next use the model to study the effect of NH3 permeability on ammonia transport. We compare PNH%= 3.1x 10-s cm/s measured by Good (1988) in the thick ascending limb versus PN~~ = 6 x lo-’ cm/s measured by Garvin et al. (1987) in the proximal tubule. We study this both in the presence (k-1 = 49,000 3-r) and absence (k-1 = 49 3-r) of luminal carbonic anhydrase and with and without H+ secretion. Table 4 shows the net total ammonia flux JNH3+NH+ and the .
’ 1.1 1
0 See text for explanation
Proc. 7th Int. Co& on Mathematical and Computer Modelling
net ammonia flux JmJ under these conditions, where Jk = /Jkdx / 27rpL. A permeability of 3.1 x 10-s is designated as “low” in the table, and 6 x lo-’ is designated as “high”. Note that when carbonic anhydrase is present in the lumen the net total ammonia flux is relatively insensitive to increased ammonia permeability, although net ammonia absorption increases substantially when the NH3 permeability is increased. This occurs with and without H+ secretion, and is due to large initial NW, reabsorption (peak J,~H~= 5.1 pmol/mm/min) in the MTAL that exceeds NH: reabsorption. In the absence of luminal carbonic anhydrase, not only is the initial NH3 reabsorption in the MTAL reduced when PNH3 is increased, but there is a substantial backflux of NHs. In the absence of proton secretion, an alkaline disequilibrium pH is generated (see Figure 7) that serves to block NH3 backflux.
Fig. 7. Disequilibrium pH versus tubule length is shown for a control simulation and for (high) P,wr = 6 x 1O-2 cm/s under various experimental conditions.
CONCLUSIONS We have described a time-dependent model of lumen acid-base balance that includes reactive and nonreactive ions, buffered and unbuffered reactions, volume flow and hydrostatic pressure. Steady-state solutions have been obtained for an initial value problem with anatomic parameters and transport coefficients measured in the thick ascending limb of Henle and with specified interstitial profiles. The model has been validated; a control experiment has been simulated, and a physiological question has been posed and investigated. We have shown that, even if the NHs permeability is relatively large, an alkaline pH disequilibrium can block NH3 backflux. However, this mechanism will not serve to sustain net total ammonia reabsorption in the rat (with H+ secretion in the TAL). Thus this suggests a low NH, permeability in the rat TAL, since reabsorption is necessary for NH, concentration via a contercurrent mechanism.
REFERENCES Burg, M. B. and N. Green (1973). Function of the thick ascending limb of Henle’s loop. J. Physiol. 224(3), 659-668. Garvin, J. L., M. B. Burg and M. A. Knepper (1987). NH, and NH: transport by rabbit renal proximal straight tubules. Am. J. Physiol. 252 (Renal Fluid Electrolyte Physiol. 21), F232-F239. Good, D. W. (1988). Active absorption of NH2 by rat medullary thick ascending limb: inhibition of potassium. Am. J. Physiol. 255 (Renal Fluid Electrolyte Physiol. 24), F78-87. Good, D. W., M. A. Knepper and M. B. Burg (1984). Ammonia and bicarbonate transport by thick ascending limb of rat kidney. Am. J. Physiol. 247 (Renal Fluid Electrolyte Physiol. 16), F35F44. Kahaner, D. K. and D. D. Barnett (1989). Plotted solutions of differential equations (Version 6.0). Center for Computing and Applied Mathematics, NIST, Washington, D.C. Knepper, M. A., R. Packer and D. W. Good (1989). Ammonium transport in the kidney. Physiol. Rev 69 174249. U, Mejia, R. (1988). Mathematical modelling of the renal concentrating mechanism. Mathl. Comput. Modelling 11, 615-620. Rocha, A. S. and J. P. Kokko (1973). Sodium chloride and water transport in the medullary thick ascending limb of Henle. J. Clin. Invest. 52, 612-623. Magnesium Shareghi, G. R. and Z. S. Agus (1982). transport in the cortical thick ascending limb of Henle’s loop of the rabbit. J. Clin. Invest. 69 (4), 759-760. Star, R. A., I. Kurtz, R. Mejia, M. B. Burg and M. A. Knepper (1987). Disequilibrium pH and ammonia transport in isolated perfused cortical collecting ducts. Am. J. Physiol. 253 (Renal Fluid Electrolyte Physiol. 22), F1232-F1242. Stephenson, J. L., Y. Zhang, A. Eftekhari and R. Tewarson (1987). Electrolyte transport in a cenAm. J tral core model of the renal medulla. Physiol. 253 (Renal Fluid Electrolyte Physiol. 22)) F982-F997. Rubidium transWork, J. and J. A. Schafer (1983). port in rat medullary thick ascending limb. Am. Sot. Neph. Abstracts, 271.