Mathematical aspects of renal function: Reabsorption of individual solutes as interdependent processes

J. Theoret. Biol. (1966) 10, 327-335

Mathematical Aspects of Renal Function: Reabsorption of Individual Solutes as Interdependent Processes MACKENZIE

WALSER

Department of Pharmacology and Experimental Therapeutics, The Johns Hopkins University School of Medicine, Baltimore, Maryland, L1.S.A. (Received 5 January 1965, and in revisedform 17 June 1965)

Introduction In the preceding paper (Walser, 1966), mathematical models of uniform segments were studied to determine the effect of changes in water reabsorption on solute reabsorption, given the kinetics for these processes at any point in the segment. Two types of solutes were considered, those whose removal in turn alters water reabsorption, i.e. sodium salts, and those present as only minor osmotic constituents, which are affected by water reabsorption but do not themselves affect water loss. The rate of reabsorption of all solutes in the latter category decreases with decreasing rate of water reabsorption (provided that their removal is a concentration-dependent process). Therefore, a comparison of the simultaneous clearances of two or more reabsorbable solutes as water reabsorption is experimentally varied (e.g. by osmotic diuresis) will reveal a relationship between the excretion fractions of the two substances. The purpose of the present study was to analyze the form of relationship between simultaneous clearances which results from different types of kinetics of solute reabsorption. Variations in clearances induced by altering the plasma concentrations of reabsorbable solutes are also examined in the same light. The symbols, kinetic equations, and methods of integration were the same as in the accompanying paper (Walser, 1966). Results EXCRETION

FRACTION OF

OF ANOTHER

ONE

SOLUTE

IN

REABSORBED

RELATION BY

SIMILAR

TO

If both solutes are reabsorbed by first-order kinetics, dz 7 dx=-5 327

EXCRETION

KINETICS

FRACTION

328

M.

WALSER

dz’ z=-%

z’

(2)

then dz -=-- a” dz’ 2 a”’ 2’

(3)

and (4)

Thus the slope of the straight line on log-log paper obtained by plotting z/z0 against z’/zb gives C/g. A somewhat more general case was presented previously (Walser & Robinson, 1963). Similarly, if both solutes share the same pump, an identical result is obtained. The rates of reaction are (Dixon & Webb, 1958): dz a -=dx l+b/y+by’/b’y dz’ a’ -=--(6) dx 1 + b’/y’+ b’y/by’ ’ since, z = yu and z’ = y’v, dz ab’ dz’ -=--; (7) Z

a’b z’

integrating, z/z0 = (z’/zp-‘b.

(8)

These reIationships are appIicable however the water is reabsorbed in the same segment. If two solutes are reabsorbed passively in a segment in which water reabsorption is linear with length, their simultaneous excretion fractions will be given by equation (22) of the accompanying paper (Walser, 1966) with different values of c. Surprisingly, these quantities are also related by a power function, although not exactly so. Table I lists (log z/z,)/(logz’/zb) for u/u,, = O-01, O-1, and 0.3, for various values of the permeability constants c and c’. This ratio is close to the ratio c/c’ but not equal to it, as may be seen. When the permeability constant for et&x exceeds that for inffux, it can be shown that this power function interdependence is followed even more closely. Indeed, as the ratio of the constant for influx to that for efflux approaches zero the interdependence approaches that shown in equations (1) to (4).

If flow decreases exponentially with length (equation (9) of the previous paper) and solute reabsorption is passive (equation (1) with c = c’), the

INTERDEPENDENCE

IN

THE

REABSORPTION

TABLE

OF

329

SOLUTES

1

Excretion fractions of two solutes, z/z, and z’/zb, and log (z/z,)/log (z’/zb), for three different values of out$ow, v/vO. Solute reabsorption passive with permeability constants c and c’; water reabsorption linear with length. Note that the log ratio is nearly independent of frow and similar in magnitude to c/c’. c

c’

v/v0

z/z0

z’jz’o

0.2

0.1

0.01 0.1 0.3

0.4917 0.7422 O-8725

0.6975 0.8585 0.9323

1.969 1.955 I.946

2 2 2

1.7

0.1

0.01 0.1 0.3

0.0234 0.1980 0,4724

0.6975 0.8585 0.9323

10.418 10,620 IO.700

17 17 17

0.4

0.2

0.01 0.1 0.3

0.2543 0.5669 0,7727

0.4917 0.7422 0.8725

1.929 1.904 1.891

2 2 2

1 .o

0.2

0.01 0.1 0.3

0.0550 0.3032 0.5822

0.4917 0.7422 0.8725

4.086 4.003 3.967

5 5 5

I.1

1.0

0.01 0.1 0.3

0.0461 0.2802 0.5611

0.0550 0.3032 0.5822

1.061 1.066 1.068

1.1

0.01 0.1 0.3

0.0234 0*1980 0.4724

0.0344 0.3032 0.5822

1.294 1.357 1.386

1.7 1.7 1.7

I.7

1.0

TABLE

(log z/zo>/(log

z’iz’d

C/d

;:;

2

Excretion fraction of solutes reabsorbed passively from a segment in which flow falls exponentially with length, in relation to flow. Values calculatedfor 6 different degrees of permeability, c, and for 10 values of outflow/inflow, v/vO. c v/v. 0.1 0.2 0.4 1.0 1.1 1.7

= 0.01 0.140 0.029 0.012 o-01 1 o-01 1 0.010

0.02

0.03

0.05

0.327 0.121 0.035 0.022 0.022 0.021

0449 0.215 0.072 0.034 0.034 0.032

0.596 0.369 0.166 O-064 0.062 0.056

0.1

0.2

0.3

0.5

0.75

0.754 0.579 0,364 0.160 0.150 0.122

0.866 0.756 0.593 0.357 0.337 0.270

0.912 0.837 0.717 0.510 0,489 0.409

0.959 0.922 0.858 0.729 0.713 0647

0.985 0.971 0.946 0.890 0.883 0.850

330

M.

WALSER

relationship between solute excretion and flow is as shown in Table 2. These results differ somewhat from those obtained when flow is linear (Table 1). However, when the simultaneous excretion fractions of two such solutes are compared, they are often found to be constant powers of one another, and these exponents have very nearly the same values as those shown in Table 1. Specifically, deviations from this pattern are seen only when the outflow concentration approaches the inflow concentration, and as indicated in Table 2, this is the case for larger values of the permeability constant or for very low or very high values of outflow. Thus the observation that clearances of two similar solutes ale interdependent in this manner is consistent with several possible kinetic mechanisms. This relationship is followed very closely by certain related substances, such as strontium and calcium (Walser & Robinson, 1963) iodide and chloride (Walser & Rahill, 1965a), and bromide and chloride (Walser & Rahill, 1966). On the other hand, the simultaneous clearances of two solutes reabsorbed by independent and partially saturated pumps do not follow this pattern. In a segment in which water reabsorption is linear with length, the excretion fractions of solutes whose inflow concentration is at least as great as the Michaelis constant are approximately linear functions of flow, as shown in the preceding paper. It follows that their excretion fractions are linear functions of one another: 4% = 4 + dh z’/zb = @+g’u/uo. Eliminating

(9) (10)

V/V,,

(11) z/z0 = (4 - &s/s’> +(s/s’>z’/zb. A comparison of the simultaneous clearances of two similar substances may therefore indicate whether they are reabsorbed by independent (and at least half-saturated) pumps or not. EXCRETION

RATE

OF

SOLUTE

AS A FUNCTION

OF PLASMA

CONCENTRATION

For most substances which are filtered and reabsorbed, the rate of reabsorption is neither independent of the concentration presented in the glomerular filtrate nor proportional to it. The relationship between reabsorptive rate and inflow concentration will not follow simple Michaelis-Menten kinetics even in a water-impermeable segment, as pointed out by Wilbrandt (1954), although most authors have continued to apply them. In this simple situation, which might apply to the distal tubule during water diuresis, the integrated Michaelis-Menten equation applies, as pointed out above. The reciprocal of the local reabsorptive rate, (dz/dx)-‘, is a linear function with

INTERDEPENDENCE

IN

THE

REABSORPTION

OF

SOLUTES

331

slope b/a of the reciprocal concentration, in accord with the usual Lineweaver-Burk plot, as seen by rearranging equation (4) of the previous paper as follows : (dz/dx)- 1 = l/a + h/uy. (12) When the reciprocal of the integrated rate is plotted in the same fashion against the reciprocal of inflow concentration, as in Fig. 1, nearly straight

01

2.0 b/v,

FIG. 1. Local solute reabsorptive rates (----) and integrated reabsorptive rates (---), plotted as reciprocals against reciprocal inflow concentration, for various values of the pump parameters a and b, in a water-impermeable segment. The integrated rates also yield nearly straight lines but the slopes of these lines are all approximately the same 1-b) unless a < b. Rate of inflow was held constant but inflow concentration, y,, was varied.

lines are also obtained, but the slopes are now given more nearly by b, and are relatively insensitive to u. Since these lines are not straight, the value of a cannot be obtained by extrapolation. However, whenever a > b the slope of a plot of (zO-z)-” against y; * gives b approximately, and estimates of a can then be obtained directly from the integrated Michaelis-Menten equation : bo,lnz/z,+z-z,

=--as

(13)

obtained from integration of dz/dx = - a/(1 +b/y) with y = z/uO. This graph also serves to emphasize that much higher inflow concentrations (relative to the Michaelis constant b) are required to achieve nearly maximal reabsorptive rates in such a segment. Inflow concentrations must exceed b by ten-fold before reabsorptive rate comes within 10% of the T,,,, a.

332

M.

WALSER

In an isosmotic segment in which the fraction of unreabsorbable solute is negligibly small throughout, sodium concentration remains constant and sodium reabsorptive rate is given directly by the appropriate kinetic equation with y a constant. In such a segment, or in any other type of segment in which flow loss is linear with length, reabsorption of other solutes follows one of the systems of differential equations already presented. For first-order kinetics z/z0 = (v/uo)d’(“o-“) . (14) Thus the fraction reabsorbed, (1 -z/zJ, is constant for any given flow but is given by 1 - (u/u$‘(“~-“) instead of by e -d’“o as would be the case in a waterimpermeable segment. The relationship between inflow concentration and reabsorptive rate for pump kinetics has been discussed by Burgen (1956). EXCRETION

RATE

OF ONE

CONCENTRATION

REABSORBABLE

SOLUTE

OF ANOTHER

IN RELATION

TO THE PLASMA

LESS REABSORBABLE

When a poorly reabsorbed solute is substituted for a well reabsorbed one in the glomerular filtrate, the clearance of both often increases. The augmented salt excretion which follows administration of a nonreabsorbable solute (even when given as an isosmotic solution), considered previously (Walser, 1966), may be viewed as a special case of this effect. Another wellknown example is the chloruretic effect of substituting a less well-reabsorbed monovalent anion such as nitrate for chloride in the glomerular filtrate. If two solutes, z and z’, are reabsorbed by first-order kinetics in the concentration ranges under consideration, and if the sum of their concentrations is fixed at some value p, dzldx = - Cz/v

(15)

dz’ldx = - a”‘z’/v

(16j

(z+z’)/u

= p

(17)

then it can readily be shown that, at x = 1, z - 20 + (6/a”‘)&[(z/z,>“““-

11 = - a”p.

This equation gives z implicitly as a fraction of the filtered loads of the two solutes. If more than two solutes are present, the solution is similar providing that the sum of the concentrations of all of the solutes is fixed. If unreabsorbable solute is also present, m In z/z0 appears as an additional term. In the proximal tubule, this model might describe such effects adequately, since the sum of the solute concentrations is fixed by the plasma osmolality.

INTERDEPENDENCE

IN THE REABSORPTION

OF SOLUTES

333

FIG. 2. The effect of the substitution of a poorly reabsorbed solute, z’, for a better reabsorbed solute, z, in the glomerular filtrate, on the excretion rate of the latter, when both are reabsorbed by Brst-order kinetics and the sum of their concentrations in the tubular fluid is constant. In this example the ratio of the first-order rate constants, &/Z’, is 5. Nearly identical curves are obtained when both solutes are reabsorbed passively or when they are both transported by a common pump.

Figure 2 shows solutions to this equation for G/i?’ = 5, and p = 1. When excretion, z, is small in the absence of z’, a considerable increment in z is produced by substituting z’ for z in the inflow. When z is large in the absence of z’, substitution of z’ for z in the inflow fails to increase excretion of z at all. When two solutes are reabsorbed passively with the constants for efflux exceeding that for influx, nearly identical curves are obtained. Similarly, two solutes reabsorbed by a common pump mechanism may exhibit the same relationships. Independent pumps, however, do not. Substitution of a less well reabsorbed solute under these conditions leads to little or no increase, and more generally a decrease in the excretion rate of the solute more readily reabsorbed. solute

Discussion

Although there have been several studies of simultaneous clearances of two or more solutes in which empirical relationships have been formulated to describe the results, the present work (and the preliminary efforts by Walser & Robinson, 1963) represent the first attempts to derive integrated kinetic equations appropriate to this type of observation. The results have in one sense been disappointing in that many types of reabsorptive kinetics can lead to single type of relationship between clearances; furthermore, this power function interdependence seems to apply (with varying precision)

334

M.

WALSER

to a large variety of different solutes (Walser & Robinson, 1963; Walser & Rahill, 19650, b, 1966; Rahill & WaIser, 1965). Further study may confirm the prediction that solutes reabsorbed by independent, partiallysaturated pumps show a different type of interdependence. If so, the utility of this type of analysis will be increased. For two solutes reabsorbed by one of the several mechanisms which lead to the power function, z/z0 = (z’/z$, it is obvious that the proximal portions of the segment account for most of the reabsorption of the more readily transported solute, while the more distal portions may account for the major portion of the reabsorption of the other. Also, a small difference in the tubular fluid-to-plasma concentration ratios of two solutes in the first half of a uniform segment is consistent with a very great difference at the end, if reabsorption of either solute is nearly complete. Change of electrical potential with length may also be considered. At any point in the tubule, the rates of removal of cations and anions must be equal. In the kinetic equations describing the reabsorption of each cation and anion, electrical potential appears as a factor of transtubular chemical gradient in the passive flux component. In the terminology employed herein the constant c’ is replaced by ce* EFiRT, where E is the transtubular potential difference, and the sign depends upon the charge of the ion. By setting the anion efflux equal to the cation efflux, it becomes apparent that there is only a single possible value for electrical potential given the concentrations of anions and cations and the kinetic equations for their reabsorption. An equation for potential as a function of length can be derived in this manner. A system of differential equations incorporating such an expression shows the expected increase in transtubular potential downstream on replacing a permeant anion with a less permeant one, and can be employed, for example, to illustrate the biphasic effect of sulfate infusion on chloride excretion. The reIationship shown in equation (18) may provide a general model for the chloruresis resulting from infusion of sodium salts of less permeant anions. The natriuresis which results from infusion of the salts of cations less well reabsorbed than sodium (lithium, for example) may represent a similar instance, and may imply that independent pump mechanisms for these cations do not exist. However, in such experiments the foreign ion is not merely substituted for the native ion in the glomerular filtrate; extracellular fluid volume is usually increased too. Consequently, the rate of glomerular filtration or the hormonal factors regulating tubular reabsorption may change, invalidating the results. The type of relationship exemplified by equation (18) could best be studied by removing chloride (or sodium) as the foreign ion (e.g. bromide or lithium) is administered, keeping extracellular fluid volume constant. Few experiments of this type have yet been reported.

INTERDEPENDENCE

IN

THE

REABSORPTION

OF

335

SOLUTES

This work was supported by a U.S. Public Health Service grant (A-2306) and by the U.S. Public Health Service Research Career Award Program (GM-K3-2583). The computer work was done in the Computing Center of The Johns Hopkins Medical Institutions, which is supported by a U.S. Public Health Service grant (FR-00004). I am greatly indebted to Dr I. M. Weiner for many helpful discussions during the course of this work; and to Drs R. B. Kelman, K. L. Zierler and L. G. Wesson, Jr, for valuable criticisms of the manuscript. Miss E. Kirstein and Mr J. C. Finck helped with the programming. REFERENCES BURGEN, A. S. V. (1956). Can. J. Biochem. Physiol. 34, 466. DIXON, M. & WEBB, E. C. (1958).“Enzymes”.NewYork: AcademicPress. RAHILL, W. J. & WALSER, M. (1965). Am. J. Physiol. 208, 1165. WALSER, M. (1966). J. Theoret. Biol. 10, 307. WALSER, M. & RAHILL, W. J. (1965a). .Z.din. Invest. 44, 1371. WALSER, M. & RAHILL, W. J. (1965b). Am. J. Physiol. 208, 1158. WALSER, M. & RAHILL, W. J. (1966). C/in. Sci., in the press. WALSER, M. & ROBINSON, B. H. (1963).In “Transfer of Calciumand StrontiumAcross

BiologicalMembranes”(Wasserman, R. H., ed.).New York: AcademicPress. W. (1954).In “Active Transportand Secretion”,Symp. Sot. exp. Biol. N.Y. 8, 136.

WILBRANDT,

22