Proc. Nail. Acad. Sci. USA Vol. 82, pp. 1563-1567, March 1985 Physiological Sciences Dissociation from albumin: A potentially rate-limiting step in the clearance of substances by the liver (drug binding/organic anions/bilirubiu/transport/models) RICHARD A. WEISIGER Department of Medicine and the Liver Center, University of California, San Francisco, CA 94143 Communicated by Rudi Schmid, August 20, 1984
ABSTRACT The hepatic uptake rate for certain albumin- tions. Most traditional models assume that equilibrium exists bound drugs and metabolites correlates poorly with their equi- between bound and free ligand within the hepatic sinusoids. librium unbound concentration in the plasma, suggesting that Recent evidence, however, suggests that the rate ofdissocia- binding equilibrium may not always exist within the hepatic tion of some ligands from albumin may be insufficient to sinusoids. Currently available models for the uptake process maintain equilibrium when hepatic uptake is rapid (3, 6). To assume binding equilibrium and, thus, cannot be used to in- address these important issues, a new formulation of the tra- vestigate this possibility. This report presents a more general ditional uptake model has been developed that treats bound model that treats plasma-bound and free concentrations sepa- and free ligand as separate plasma compartments. The re- rately. A solution is provided that specifies the hepatic uptake sulting analysis indicates that the kinetic behavior that has rate as a function of the total plasma concentrations of the previously been attributed to an albumin receptor can also transported substance and of binding protein and the rate con- be produced by the traditional uptake model. In addition, stants for influx, efflux, elimination, association, dissociation, this approach provides numerous insights into the kinetic be- and flow. Analysis of this solution indicates that hepatic up- havior expected for the traditional model, which should take may be limited by the rate of plasma flow, dissociation prove useful in interpreting the results of pharmacokinetic from the binding protein, influx into the liver, cellular elimi- and transport studies. nation, or any combination of these processes. The affinity and concentration of the binding protein strongly influence METHODS which of these steps are rate-limiting in any given case, and For this analysis, the liver is treated as a collection of equiv- binding equilibrium exists within the hepatic sinusoids only for alent tubular sinusoids, each of which is divided into N axial binding protein concentrations greater than a specified value elements (see Fig. 1). Within each element plasma-bound (the ratio of the uptake and association rate constants). The and free ligand are treated as separate compartments that precise conditions under which each step is rate-limiting and exchange according to the rate constants for complex disso- the kinetic behavior expected when two or more steps mutually ciation (r2) and formation (rg A). The concentrations of free limit uptake are provided. The results are compatible with ligand (L), albumin-ligand complex (C), and free albumin (A) previously reported data for the uptake of certain albumin- entering the first element are the equilibrium values of the bound ligands such as bilirubin, and they offer an alternative bulk plasma, while each subsequent element receives the ef- to attributing these kinetics to the presence of an albumin re- fluent of the element which preceeds it. The net rate of up- ceptor. take (or removal) for a single element is found by solution of the steady-state mass balance equations (see Appendix). The The degree to which a drug or metabolite is bound to albu- rate for the entire sinusoid is found by summation of the min in plasma is recognized as an important determinant of rates for all elements. When implemented on a small com- its rate of removal by the liver and other tissues. Indeed, it puter, this approach allows convenient plotting of the pre- has long been assumed that only the unbound form is avail- dicted net uptake velocity as a function of any one or more of able for uptake (1). This assumption is not necessarily valid, the model parameters. however, because the bound substance always retains some The conditions under which each step in the uptake proc- free energy (2), which makes it a potential substrate for an ess (i.e., plasma flow, dissociation, influx, and elimination) uptake process. Recent reports have, in fact, suggested that is rate limiting were determined by inference from the gener- for several albumin-bound substances (subsequently re- al solution. The equation that governs uptake when each ferred to as ligands), the rate of hepatic removal correlates step is rate-limiting was first determined, and then the condi- much more closely with the bound than with the equilibrium- tions under which the general solution simplifies to that free concentration (3, 4). To account for these and related equation were found (see Appendix). The resulting condi- data, it has been proposed that uptake may occur not only tions are those for which that step is solely responsible for from the free ligand pool, but also (less efficiently) from the determining the uptake rate. bound ligand pool by a mechanism involving transient bind- ing of albumin to receptor sites on the cell surface (5, 6). THEORY According to this model, the latter pathway predominates This is a linear model (i.e., transport and elimination are as- for tightly bound ligands such as bifirubin because of their sumed not to be saturable for the range of ligand concentra- very small unbound concentrations in plasma. tions used). Saturation of the albumin-binding sites at higher Although the albumin-receptor model is attractive, it is im- concentrations of ligand is, however, permitted. This model portant to consider whether the traditional view that uptake extends existing models to cover conditions for which bind- depends only on the unbound ligand concentration at the liv- ing equilibrium may not be present. When a single sinusoidal er cell surface might not also accommodate these observa- element is used (N = 1), the model is structurally similar to the "well-stirred" or "venous equilibrium" model, while for The publication costs ofthis article were defrayed in part by page charge large values of N it is similar to the "parallel tube" or "sinu- payment. This article must therefore be hereby marked "advertisement" soidal" model (7). The conditions under which each step is in accordance with 18 U.S.C. §1734 solely to indicate this fact. rate-limiting were derived for N = 1 and thus, strictly speak-
1563 Downloaded by guest on September 26, 2021 1564 Physiological Sciences: Weisiger Proc. Nad Acad Sci. USA 82 (1985) ing, apply only to the well-stirred model. However, the num- which governs the rate of hepatic removal of total ligand ber ofelements used had little effect on the resulting velocity from the sinusoid under equilibrium conditions (sec'1). except when the single-pass extraction of ligand exceeded 80%-90%6. Ten elements were arbitrarily used for all graphic RESULTS simulations in this study, although results in each case were Four discrete steps may limit the net rate of ligand uptake at effectively independent ofthe number ofelements for N > 3. steady state: plasma flow, dissociation from albumin, influx, This same basic approach has been used previously to inves- and elimination. The conditions under which each of these tigate the effects of slow rates of dissociation, although with steps is rate-limiting are given in Table 1. a different emphasis (8). Flow-Limited Uptake. Uptake is exclusively flow-limited The solution for a single sinusoidal element at steady state whenever the uptake rate is equal to the product of the flow is given by Eq. 1, where V is the net hepatic uptake (remov- rate and the total ligand concentration. For the general solu- al) velocity (see units below), Ltin is the total entering ligand tion to simplify to this equation, the rate constant for flow concentration, At is the total entering albumin concentration must be much smaller than both the rate constants for total (assuming one binding site per albumin molecule), Li is the ligand uptake and fpr dissociation (Table 1). These condi- entering free ligand concentration, and S is a composite rate tions assure that neither uptake nor dissociation can be rate- constant that incorporates the rate constants for influx (kj), limiting. efflux (k2), and elimination (k3), and that governs the rate of Dissiation-Limited Uptake. Uptake is limited by the rate removal of free ligand from the sinusoid. Q, r1, and r2 are the of dissociation from Ialbumin whenever the uptake rate is rate constants for flow, association, and dissociation. To equal to the product of the dissociation rate constant and the simplify subsequent analysis by allowing direct comparison bound ligand concentration. The first condition requires that between rate constants, V is expressed per unit sinusoidal flow must be large enough to avoid flow-limited uptake, but volume (mol sec, 1cm-3), while Q is expressed as the rate small enough that dissociation remains the primary source of at which plasma within the element is replaced by flow free ligand within the sinusoid. The second condition speci- (sec-1). r, has units of sec-' M-1; all other rate constants fies that the albumin concentration must be large enough have units of sec'1.
- - + + + - - + + + + r(Lon At) (r2 Q)(1 S/Q) V[ri(Lt,in At) (r2 Q)(1 S/Q)]2 4r1(1 S/Q)(r2 Ltin + Q-Lin) [1] 2r1(1/Q + 1/S) Where that most ligand is bound, but small enough that binding c= k3 [2] equilibrium is not present within the sinusoid (see below). Az2 + Az3 Intrinsically Limited Uptake. Uptake is limited by the in- trjnsic capacity of the liver to remove ligand from the sinu- From Fig. 1, it is evident that ligand in the bulk plasma soid whenever the uptake rate is equal to the product of the must pass through several intermediate compartments be- intrinsic uptake rate constant (S) and the equilibrium free fore it can be eliminated by the liver. A small number of ligand concentration (Le). Two cases exist-. When k2 is parameters govern exchanges between these compartments, <
plasma and cytoplasm, and S simplifies to the product of the C) shows the effect of reducing r2 until it is 1, which is equivalent to ever the single-pass extraction of ligand is measurably large, plasma flow is at least partially rate-limiting to uptake. Like- A > r2/rl. [3] wise, whenever the rate of efflux from the liver at steady state is a significant fraction of influx, elimination is at least upper free albumin concentra- partially rate-limiting to uptake. The limit of this region is the tion necessary to produce binding equilibrium within the si- Since the free albumin concentration (A) and the affinity nusoid. Higher values of A favor equilibrium because they of the albumin binding sites (Kd) appear in many of the con- reduce the removal rate (by reducing the free ligand concen- ditions in Table 1, these parameters may greatly influence tration) while not affecting the rate of dissociation. Since which step limits uptake. Several cases are possible. equilibrium requires that the rates of ligand binding and dis- [S << Q]. The simplest situation occurs when the rate constant for intrinsic hepatic removal is much smaller than sociation be effectively equal, the rate at which free ligand is cleared by the liver (SOL) must be small with respect to the the rate constant for flow. In this case, the rate at which free ligand flows into the sinusoid is always much greater than rate of association (r1 -A L) the rate of removal by the liver, and uptake is intrinsically limited for all values of A and Kd. Fig. 2 (Case A) demon- S-L <
CaseA: S<< Q CaSe B: C «<< S << r2 Case C: Q << Sand r2<< S n rho 41 .0r^
r;O 0.016 0.75-
CL Intrinsically- 0) Intrinsically-limited Flow-limited limited 0.50 1 \ 1 e) 14 Co. U)d) 0.0 CY) -C 0.0112 C/) 0.25
ol 1 10-2 10° 102 104 106 10-2 100 102 1 04 106 102 A/4d AlKd A/Kd
FIG. 2. Effect of the degree of albumin binding (A/Kd) on the fraction of ligand that is extracted by the liver in a single pass. Case A [S << Q], uptake is limited by intrinsic uptake capacity of liver for all binding levels. Case B [Q << S << r2j, uptake is flow-limited for lower levels and intrinsically limited for higher levels. Case C [Q << S and r2 << S] is similar to case B except that dissociation-limited uptake may occur at intermediate levels. In'these simulations, A was varied and Kd was held constant. Parameter values are as follows: Case A: S = 0.01 sec1, Q = 0.33/N sec-1, r2 = 1000 sec1, r1 = 1010 sec-1-M-1, Lt in = 10-11 M, and N = 10. Case B: same as case A except that S has been increased to 100 sec-1. Case C: same as case B except that both r, and r2 have been divided by 104. Downloaded by guest on September 26, 2021 1566 Physiological Sciences: Weisiger Proc. Nad Acad Sd USA 82 (1985) These conditions are equivalent to those found previously by exist. For the first "equilibrative" class, the physiologic con- inference from the general solution (Table 1). centration of albumin or other binding protein is >S/rl (due Effect of Varying Ligand at Constant Albumin. Simulations to low values of S, large values of rl, or both). For this class, based on Eq. 1 indicate a linear response between the total the plasma within the hepatic sinusoids will behave as a sin- ligand concentration and the uptake rate, providing only that gle compartment and the uptake rate at a steady state will the binding capacity of the albumin is not Significantly satu- reflect the equilibrium free concentration of the ligand. War- rated (data not shown). Uptake is proportional to the total farin appears to belong to this class (9). The second, "non- ligand concentration for all values of all parameters, regard- equilibrative" class includes those ligands for which physio- less of which uptake step is limiting. logic binding protein concentrations are insufficient to main- Effect of Varying Both Ligand and Albumin. For A > Kd, tain binding equilibrium within the sinusoids. For these the equilibrium free ligand concentration depends only on compounds, the sinusoidal plasma will behave as separate the molar ratio of L to A, and not on their absolute concen- bound and free compartments. Since dissociation is rate-lim- trations. Thus, simultaneous variation of At and Lt,in at a iting, uptake will reflect the bound rather than the equilibri- fixed molar ratio will always cause the uptake velocity to um free concentration. This type of behavior has been ob- approach a limiting value (S-L.). For values of S too small to served in the perfused rat liver for bilirubin, sulfobro- produce flow- or dissociation-limited uptake, the limiting up- mophthalein, long chain fatty acids, taurocholate, and rose take rate is reached for A >> Kd (Fig. 3, Case A). When S is bengal (3-6, 10-12). However, the albumin concentrations larger, the albumin concentration required to produce intrin- used in these studies were typically lower than physiologic, sically limited uptake will be greater. The saturation-like be- and it remains to be determined if dissociation of any of havior that results is due to the transition from flow-limited these ligands is rate-limiting to uptake under physiologic to intrinsically limited uptake (Case B) or from dissociation- conditions. limited to intrinsically limited uptake (Case C) and is not pro- Available experimental data are frequently insufficient to duced by actual saturation of a receptor site or transport determine which step in uptake is rate-limiting. Although it is mechanism. usually possible to determine if flow is limiting from the measured or calculated single-pass extraction of ligand, it is more difficult to discriminate among the other uptake steps, DISCUSSION because the rate constants governing these processes are The current study indicates that at least four different steps rarely known. The data are most difficult to interpret when may limit the rate of hepatic uptake of protein-bound drugs they have been obtained at a single binding protein concen- and metabolites. Emphasis has been placed on the role of tration, since a linear relationship between net uptake veloci- protein binding in determining the rate-limiting step, since ty and total ligand concentration is expected regardless of failure to fully consider this factor is a potential source of which steps are limiting. Even if saturation of a membrane confusion within the literature. In particular, studies that in- carrier or elimination process can be demonstrated at higher volve variation ofthe albumin concentration may be difficult ligand concentrations (indicative of intrinsically limited up- to interpret if the rate-limiting uptake step is changed. take), it cannot be assumed that this same process remains Although sinusoidal bound and free ligand are treated as rate-limiting at lower concentrations of ligand, which are of- separate compartments in this analysis, in many cases this is ten more physiologic. Saturation necessarily lowers the unnecessary. However, the current study indicates that the effective value of S and may, thus, artifactually convert sinusoidal plasma will behave as a single compartment only flow- or dissociation-limited uptake to intrinsically limited when the free binding protein concentration is >S/rl. Since uptake. S and r1 vary among ligands, two classes of ligands must Several considerations limit the predictive value of this analysis. First, for simplicity, albumin has been assigned a single class of non-interacting binding sites despite the fact that it often displays multiple sites of variable affinity. Al- though an exact solution for the behavior of the traditional 0) model under these more general conditions is not available, numerical simulations suggest that the effect is to broaden .E the range of albumin concentrations over which transitions between different uptake steps occur (unpublished data). The second major limitation is that values for several of .E0) the parameters necessary to solve Eq. 1 are poorly known. For a variety of ligands, including bilirubin, oleate, tauro- 0 cholate, and sulfobromophthalein, the single-pass extraction 10 o0>~ by the perfused rat liver in the absence of albumin is 0.95- .5 0v 0.998 (unpublished data), suggesting that S must be at least 1-2 sec1. If, however, the sinusoids are ofunequal length or are perfused unequally, S could be much larger. Estimated values for r2 for several of these ligands range from 0.003 to 0.04 sec-1 (13-16). Since these values are well below the minimum estimate for S, Inequality 7 is satisfied and a range Albumin, mM of albumin concentrations must exist for these ligands for which uptake is dissociation-limited. 0 0.01 0.02 0.03 0.04 0.05 Estimates for r1 are less certain. Early events in associa- Ligand, mM tion appear to be diffusion-limited, with rate constants for several tiphtly bound ligands ranging from 1-40 x 106 FIG. 3. Effect of varying both ligand and albumin concentrations M-1 sec- (13, 17-22). However, the rate-limiting step in at a fixed molar ratio (0.1). Uptake approaches a limiting velocity in may not each case as albumin concentration becomes large enough to pro- complex formation be this initial interaction, but duce intrinsically limited uptake. Parameter values are the same as rather the much slower process of conformational relaxation for Fig. 2. Limiting velocities are as follows: Case A, 0.11 pmol- of the albumin molecule (typically on the order of 2 sec-1), sectlcm-3; Cases B and C, 1.1 nmol sect cm-3. which is believed to increase the affinity of the interaction Downloaded by guest on September 26, 2021 Physiological Sciences: Weisiger Proc. NatL Acad Sci USA 82 (1985) 1567
(13, 17-21). Because of this large uncertainty in the value of -b + Vb2 - 4a-c r1, it is not possible to determine from inequality 5 whether V 2= [A5] physiologic albumin concentrations are sufficient to main- 2-a tain sinusoidal binding equilibrium for these ligands. "Saturable" uptake kinetics similar to those shown in Fig. while the equation for each special case has the form: 3 have recently been found for long chain fatty acids (3), bile acids (4), bilirubin (11), sulfobromophthalein (10, 11), and V = d [A6] rose bengal (12). It has been proposed that these kinetics reflect the presence of sites on the liver cell surface that tran- where a, b, c, and d are expressions in At, Ltj, Li,, rl, r2, kj, siently bind albumin and catalyze the transfer of ligands k2, k3, and Q. The conditions for simplification were found from albumin to the cell. Such a mechanism has been inde- by determining the conditions under which these equations pendently supported by the demonstration of saturable bind- are approximately equal. Squaring the resulting expression ing of albumin to liver cells, and it can be shown that some and rearranging gives the general condition for simplification form of catalyzed dissociation must exist to explain the ob- + 4'a-b-d + 4-ac + b2- served rates of uptake if the published dissociation rate con- 4-a2'd2 b2. [A7] stants are correct (5, 6). Nevertheless, the traditional model The specific conditions were found in each case by expan- presented here remains a viable alternative to the albumin- sion of Eq. A7 and finding the conditions necessary to elimi- receptor model. Careful measurement of the rate constants nate the appropriate terms. In some cases, the resulting con- for dissociation of these ligands from albumin under physio- ditions were combined to give the conditions listed in Table logic conditions appears to be the most promising method for 1. For example, the second condition for dissociation-limited discriminating between these alternative uptake models. uptake combines the simple conditions, A << S/rr, A >> Kd and r2<< S, each of which was useful in eliminating APPENDIX terms. I thank Dr. Vojtech Licko for valuable consultation on the mathe- Derivation of Net Uptake Rate for a Single Sinusoidal Ele- matical derivations. This work was supported by Research Grant ment. At steady state, the values for L, C, and L' in each AM-32898 by the National Institutes of Health and by a Hartford element are constant: Foundation Fellowship. 1. Von Leeuven, W. S. (1924) J. Pharmacol. Exp. Ther. 24, 13- L = Q(Li, - L) 19. - k1L + k2q3*L' - r1 A L + r2&C = 0 [Al] 2. Tanford, C. (1981) Proc. Natl. Acad. Sci. USA 78, 270-273. 3. Weisiger, R. A., Gollan, J. L. & Ockner, R. K. (1981) Science C = Q(Ci. - C) + r1 A L - r2 C = 0 [A2] 211, 1048-1051. 4. Forker, E. L. & Luxon, B. A. (1981) J. Clin. Invest. 67, 1517- = k1 L - (k2 + k3)'j3L' = 0 [A3] 1522. 5. Weisiger, R. A., Gollan, J. L. & Ockner, R. K. (1982) Prog. of where Li, and Cin are the concentrations of free ligand and Liver Dis. 7, 71-85. complex flowing into the element, and is the ratio between 6. Ockner, R. K., Weisiger, R. A. & Gollan, J. L. (1983) Am. J. P Physiol. 245, G13-G18. the cytosolic and plasma volumes. Eq. A3 is solved for L' 7. Pang K. S. & Rowland, M. (1977) J. Pharmacokinet. Bio- and the result is substituted in Eq. Al, which is solved for C pharm. 5, 625-653. and substituted in Eq. A2. After substitution of At - C for A 8. Jansen, J. A. (1981) J. Pharmacokinet. Biopharm. 9, 15-26. and Lt jn for Lin + Cin, the result is a quadratic equation in L. 9. Levy, G. & Yacobi, A. (1974) J. Pharm. Sci. 63, 805-806. 10. Weisiger, R. A., Zacks, C. M., Smith, N. D. & Boyer, J. L. (1984) Hepatology 4, 492-501. + - A) - + + r1(1 )L2 [r,(Lt i (r2 Q)(1 )]L 11. Weisiger, R., Gollan, J. & Ockner, R. (1980) Gastroenterology Q 79, 1065 (abstr.). 12. Forker, E. L., Luxon, B. A., Snell, M. & Shurmantine, W. 0. - r2 Lti - Q-Lin = 0 [A4] (1982) J. Pharmacol. Exp. Ther. 223, 342-347. 13. Gray, R. D. & Stroupe, S. D. (1978) J. Biol. Chem. 253, 4370- 4377. 14. Svenson, A., Holmer, E. & Anderssen, L.-O. (1974) Biochim. where S is a composite rate constant equal to klgk3/(k2 + k3). Biophys. Acta 342, 54-59. Solution of this equation by the quadratic formula gives L as 15. Faerch, T. & Jacobsen, J. (1975) Arch. Biochem. Biophys. 168, a function of At, Ltin, Lin, and the relevant rate constants 451-457. (only the positive root gives valid results). Multiplying by S 16. Scheider, W. (1978) Biophys. J. 24, 260-262. yields the expression for the net uptake velocity at steady 17. Koren, R., Nissani, E. & Perlmutter-Hayman, B. (1982) Bio- state (V), and it is given as Eq. 1 in the text. chim. Biophys. Acta. 703, 42-48. Special Cases. Each step in the uptake process is charac- 18. Wilting, J., Kremer, J. M. H., Ijzerman, A. P. & Schulman, terized by a simple equation (Table 1) that represents a spe- S. G. (1982) Biochim. Biophys. Acta 706, 96-104. cial case of the The 19. Faerch, T. & Jacobsen, J. (1977) Arch. Biochem. Biophys. 184, general solution. conditions under which 282-289. the general solution simplifies to each special case were 20. Scheider, W. (1979) Proc. Natl. Acad. Sci. USA 76, 2283- found as follows. 2287. Since the general solution was found by the quadratic for- 21. Chen, R. F. (1974) Arch. Biochem. Biophys. 60, 106-112. mula, it has the form 22. Reed, R. G. (1977) J. Biol. Chem. 252, 7483-7487. Downloaded by guest on September 26, 2021