Binding, Internalization, and Intracellular Processing of Proteins Interacting with Recycling Receptors A KINETIC ANALYSIS*

We measured time-dependent concentration changes of human interferon-ana (IFN) and human tumor necrosis factor-a (TNF) bound at the plasma membrane and internalized by human lung alveolar carcinoma A649 cells in the presence of excess free ligand. Con- centration changes for these two ligands were substan-tially different. We modified our compartmental ki- netic model encompassing receptor synthesis and receptor loss (Myers, A. C., Kovach, J. S., and Vuk- Pavlovic, S. (1987) J. Biol. Chem. 262, 6494-6499) to include receptor recycling. We solved analytically the equations of three variants of the model of receptor recycling. All parameters (rate constants) were iden- tifiable when the data sets consisted of time-resolved concentrations of IFN and TNF at the cell surface and internalized by cells. By least squares fitting we derived the best fit values for the first order rate con- stants for internalization of the ligand-receptor complex, receptor recycling, turnover of free receptors, elimination of the ligand from cells, and the rate of insertion of free receptors into the membrane. The best fit to data for interactions of cells with IFN was obtained without inclusion of the term for recycling of receptors to the membrane. The simplest model includ- ing receptor recycling was necessary and sufficient for the fit to the respective data for TNF. These results demonstrate that the contribution of receptor recycling to the metabolism of the ligand and the receptor can be quantitated by compartmental modeling.

We measured time-dependent concentration changes of human interferon-ana (IFN) and human tumor necrosis factor-a (TNF) bound at the plasma membrane and internalized by human lung alveolar carcinoma A649 cells in the presence of excess free ligand. Concentration changes for these two ligands were substantially different. We modified our compartmental kinetic model encompassing receptor synthesis and receptor loss (Myers, A. C., Kovach, J. S., and Vuk-Pavlovic, S . (1987) J. Biol. Chem. 262,[6494][6495][6496][6497][6498][6499] to include receptor recycling. We solved analytically the equations of three variants of the model of receptor recycling. All parameters (rate constants) were identifiable when the data sets consisted of time-resolved concentrations of IFN and TNF at the cell surface and internalized by cells. By least squares fitting we derived the best fit values for the first order rate constants for internalization of the ligand-receptor complex, receptor recycling, turnover of free receptors, elimination of the ligand from cells, and the rate of insertion of free receptors into the membrane. The best fit to data for interactions of cells with IFN was obtained without inclusion of the term for recycling of receptors to the membrane. The simplest model including receptor recycling was necessary and sufficient for the fit to the respective data for TNF. These results demonstrate that the contribution of receptor recycling to the metabolism of the ligand and the receptor can be quantitated by compartmental modeling. Receptor recycling does not contribute to the kinetics of Type I IFN receptor in A549 cells. In contrast, recycling contributes significantly to endocytosis mediated by the TNF receptor.
Ligands such as growth factors, protein hormones, and trophic proteins interact with cellular receptors and trigger transduction of biochemical signals which elicit biological *This work was supported in part by Grants CA45312 and CA15083D2 from the National Cancer Institute, Department of Health and Human Services, by Hofmann-LaRoche, and by the Fraternal Order of Eagles Cancer Research Fund. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18  responses. These processes are usually accompanied by endocytosis of the ligand-receptor complex and by replenishment of free membrane receptors. In absence of ligand, the free receptors in the membrane maintain a steady-state concentration; this concentration is determined by the ratio of receptor insertion rate into the membrane and the rate of loss of free receptors from the membrane by internalization, shedding, or inactivation of receptors (1). The rate of internalization for any particular ligand will be determined by ligand concentration, receptor concentration, ligand-receptor affinity, rates of ligand synthesis and loss, receptor recycling (reutilization), the rate of endocytosis, and the transport capacity of the endocytotic apparatus (1).
Understanding the regulatory factors critical for interactions of protein ligands with cells is particularly important for rational manipulation of extravascular protein trafficking, as in autocrine regulation (2, 3) and for biological response modifier therapy (4). For example, "sinking" of proteins into cells by receptor-mediated endocytosis might restrict the efficiency of diffusional and convectional transport of proteins in tissues (5).
In order to resolve kinetic determinants of protein-cell interactions, we are studying interactions of proteins with cells in monolayer culture (6). Compartmental models provide means to analyze time-dependent concentrations of both ligand interacting with the cell surface and of internalized ligand. Recently we reported derivation of rate constants by computer-assisted modeling for interactions of IFN' with the nonrecycling Type I IFN receptor in human carcinoma cells in uitro (6). We now report derivation of a similar mathematical model which includes recycling of receptors. We have considered applicability of simple models that include firstorder rate constants for elementary processes in receptor metabolism. We have investigated identifiability of parameters, reported analytical solutions of the proposed models, tested these solutions on data (interactions of IFN and TNF with human alveolar lung carcinoma A549 cells in vitro), and investigated whether statistical criteria alone are adequate discriminants of nonrecycling receptors and recycling receptors. We also introduce methods for reduction in the number of free parameters per experimental point for compartmental modeling of receptor dynamics. A comparative kinetic analysis of interactions of IFN and TNF with human epithelial tumor cells in uitro will be reported separately.

MATERIALS AND METHODS
Cells-Human lung adenocarcinoma A549 cells were obtained from American Type Culture Collection (Rockville, MD). Cells were main-' The abbreviations used are: IFN, recombinant human interferonma; TNF, recombinant human tumor necrosis factor-a. tained in Dulbecco's modified Eagle's medium (GIBCO) containing 10% heat-inactivated fetal calf serum (GIBCO) at 37 "C in watersaturated 5% carbon dioxide, 95% air mixture.
Radioiodination of Tumor Necrosis Factor-Recombinant human TNF (Lot 3056-55; specific activity: 5.02 X lo7 units/mg; Genentech, San Francisco, CA) was iodinated according to a published procedure (7). A quantity of 3.7 X lo7 Bq (1 mCi) of NalZ5I (specific radioactivity: 9.3 X 10l6 Bq/mol) in 10 p1 of 0.1 M NaOH (Amersham Corp.) was added to a 1.5-ml conical polystyrene tube containing 100 pl of 0.1 M sodium phosphate, pH 7.4, and two Iodo-beads (Pierce Chemical Co.). After incubation of the tube on ice for 10 min, 100 p1 of the solution was transferred to another tube containing 10 pg (20 pl) of TNF. Incubation on ice was continued for 5 min and terminated by gel filtration on a PD-10 (Sephadex G-25) column (Pharmacia, Uppsala, Sweden) equilibrated with 0.2 M phosphate, pH 7.4, containing 0.1% gelatin. Specific radioactivity of these preparations was -5 X 1015 Bq/mol. On the average, 93% radioactivity was precipitable by 8.5% trichloroacetic acid in presence of 10% fetal calf serum as carrier protein. The biological activity of radiolabeled TNF was indistinguishable from native TNF based on lysis of murine L929 cells pretreated with actinomycin D (1 pg/ml) with 2-fold dilution series of native TNF and labeled TNF (7). lZ5I-TNF was stored at 4 "C and showed no change in binding and internalization over 4 weeks.
Measurement of Dksociable and Nondissociable Tumor Necrosis Factor in A549 Cells-Measurement of dissociable and nondissociable TNF was performed in confluent cell layers in 35-mm plastic Petri dishes (Falcon, Oxnard, CA). Briefly, experiments were initiated by collection of the conditioned media overlying the confluent cells. The medium was centrifuged at 3000 X g for 10 min at room temperature when the clear supernatant was removed. The experiment was initiated by adding '"1-TNF to the supernatant at concentrations indicated in Tables I1 and 111. After warming the medium to 37 "C, 1.5 ml was added to each of multiple plates kept at 37 "C. At times specified in the figures, triplicate plates were placed on ice. The medium was removed and the cells washed three times with 1 ml of ice-cold phosphate-buffered saline, pH 7.4, containing 10% fetal calf serum. More than 95% membrane-bound radioactivity was eluted from cells by washing twice with 0.2 M acetic acid containing 0.5 M sodium chloride, pH 3.0 (8). For determination of internalized radioactivity, acid washed cells were solubilized in 1 ml of 0.5 M sodium hydroxide. After solubilization and removal of fluid, the dishes were washed once with 1 ml of 0.5 M sodium hydroxide. This wash, combined with the solubilized cells, was assumed to contain the total amount of internalized radioactivity. Radioactivity was measured by a Beckman 4000 y-counter with 66% efficiency.
Nonspecific binding and internalization were determined in parallel experiments conducted in presence of 5.6 X lo-' M TNF. At no time did the membrane-associated radioactivity in the presence of native cytokine exceed 10% of the value measured in absence of native cytokine. The corresponding values for the internalized radioactivity never exceeded 6%. The Models and Parameter Fitting-We previously considered a three-compartment model for interactions of proteins with nonrecycling receptors under pseudo-first order conditions with respect to the concentration of the free ligand assuming first order rate constants (6). In the absence of ligand ( t = 0), this model (Model I; (R)s. The apparent fiist order recycling rate constant characterizing recycling is k, (Model I1 in Fig. 1). Equations 1 and 3 then become Under certain conditions, the (radioactive) ligand and/or its degradation products accumulate within a compartment of intracellular receptor-free radioactivity (L). This compartment is included into Model I11 (Fig. 1) described by Equations la, 2, 3b, and 4.
When ligand binding triggers delayed irreversible changes of the physical state of membrane receptors (e.g. aggregation; cf. Ref. In), a distinct pool of membrane receptors committed to internalization (LR)s,c is created. Conversion of (LR)s into (LR)s,c is characterized by the first order rate constant kc (Model 1% Fig 1, Equations la, 2b, 5, and 3d).
If the free ligand concentration is constant throughout experiments (or if the change is negligibly small), differential equations defining Models I-IV are first order and can be explicitly solved. The explicit solutions for the models are given in Appendix I? The interactive computer graphics methods and the curve fitting program were described by Myers et al. (6). The best fit parameters were obtained as values of first order rates or rate constants for all parameters: IZ.
[L], kd, ke, kh, V,, as well as k, and kc when applicable.
Equations were fitted to the means of triplicate measurements f S.E. These errors reflected variations in number of cells among dishes and the Poissonian nature of radioactive decay. When experimentally determined standard errors were smaller than predicted by the Pois-Appendices I and I1 are presented in miniprint at the end of this paper. Miniprint is easily read with the aid of a standard magnifying glass. Full size photocopies are included in the microfilm edition of the Journal that is available from Waverly Press. son distribution, we used the square root of the mean value as the standard error.
The uncertainty in best fit parameter values was determined by standard methods (13). The goodness of fit is reported by values of X; (x2 divided by the number of degrees of freedom). The significance of differences between xf values was evaluated by the F-test (13).
The best fit curves were tested for serial correlations of error terms according to Durbin and Watson (14,15).

RESULTS
In this paper IFN and TNF interactions with A549 cells have been studied as a paradigm for testing Models I-IV. A549 cells were selected because they provided easily measurable interactions both with IFN (6) and TNF.
Interactions of A549 Cells with IFN-When IFN interacted with cells under pseudo-first order conditions, both [LR]s and [LRIr depended on time (Fig. 2 in Ref. 6). [LRIsvalues reached a maximum approximately 30 min after exposure of cells to IFN and then declined to a steady-state concentration.
[LR], peak values were observed around 100 min and declined to a steady-state level. For a discussion of mechanisms of these concentration changes cf. Refs. 1 and 6.
We fitted equations of Models I-IV to our previously published data (Fig. 2 in Ref. 6). The best fit parameters for Model I (nonrecycling receptors) are given in Table I Equations of Model I1 (recycling receptors) were fitted to data with two different sets of initial parameters. With best fit parameters from Model I and k, = 0 as initial estimates, the best fit parameters for Model I1 did not differ from initial estimates, indicating a local minimum. This fit was robust and the best fit parameters did not change if single initially entered parameters were varied within 1 order of magnitude. If several initial parameters were perturbed by several orders of magnitude, another best fit was reached for Model I1 and the x: value was 6.8 compared to 11.7 for Model I? However, the value of the recycling rate constant k, was negative indicating that this set of best fit parameters represented a physically unfeasible solution. This conclusion is corroborated by the negative sign of the calculated values of [R]s = f ( t ) . Similarly unrealistic sets of best fit parameters were obtained by fitting equations of Models I11 and IV to same data (not shown). Thus, Model I yielded the optimal fit to data and physically adequate parameter values, although smaller x?
values were obtained in a fit which resulted in physically unrealistic best fit parameters.
Reduction in Number of Free Parameters-The fit of equations of Model I to data (Fig. 2 in Ref. 6) was obtained with all parameters free and without limits imposed on values o f acceptable solutions. However, when one or more parameters of the model are known a priori, they can be used as fixed parameters in the fitting procedure to reduce the number of free parameters. This manipulation results in fewer free parameters per experimental point and has been shown to result in better fits (11,16).
In order to reduce the number of free parameters, we used the equilibrium dissociation constant KD. This constant, usually derived by the Scatchard analysis of equilibrium binding data is known for many cell-protein interactions. We modified the format of parameters to include the value of &/[L] as the fixed parameter. However, the extent to which the best Note that the x? values reported in Ref. 6 should be corrected by squaring and multiplying by the number of degrees of freedom. fit parameters are sensitive to errors in the fixed parameter might limit the usefulness of the independently derived fixed parameter. Therefore, we investigated how the best fit parameters and x: value from Model I for data (Fig. 2 in Ref. 6) varied as the function of KO.

ad-Receptor Interactions
Parameters and x? values from best fits of Model I to data  maximum (k,, and kc) or Interaction of A549 Cells with TNF-The time course of TNF interactions with A549 cells in vitro is represented in Fig. 3. In the presence of TNF (1.5 X 10"' M) in culture medium, the surface-associated radioactivity, (LR)s, was maximal approximately 10 min following introduction of TNF. Then the surface-associated radioactivity decreased somewhat and increased steadily thereafter (see inset in Fig. 36). The internalized radioactivity, (LR)I, increased throughout the experiment (Fig. 3). In the course of 6 h of interaction of TNF with A549 cells, the values of [LR]s and [LRII did not reach a steady state in clear distinction to interactions of cells with IFN.

TABLE I Best fit parameters for the interaction of human interferon-cu with A549 cells
In order to determine the model most adequate for description of kinetics of interactions of TNF with A549 cells, we fitted equations of Models I to IV to data in Fig. 3. The respective best fit parameter values and the corresponding xf values are listed in Table 11.
Model I-For this model of nonrecycling receptors, the curves for [R]s, [LRIs, and [LRII were fitted to data to yield the best fit values to six free parameters (Fig. 3a).  Table 11) was significantly lower than x: for the fit of the six-parameter Model I (F = 44.8; p < 0.001). All best fit parameters for Model I1 were positive and the coefficients of variability were smaller than 15%, except for k h (Table 11).
Model 111-The absence of an apparent steady state in [LRII (Fig. 3) and the small value of k h obtained from Model I1 (Table 11) indicated the possible presence of an internalized ligand pool. Therefore, we modified Model I1 to include such a pool created by the k, process and reduced by the k h process (Model 111). The xf of 16.8 was more than twice as large as the corresponding value for Model 11. The unrealistically high value of kh and the x: value as large as for Model I indicate that Model I11 was significantly less adequate for data in Fig.  3

than Model 11.
Model IV-Association of protein ligands with specific receptors in plasma membrane is generally accompanied and/ or followed by preendocytotic aggregation of the ligand-receptor complexes in the plane of the membrane (cf. Refs. 11 and  12). Therefore, we investigated how introduction of the preinternalization step (as in the seven-parameter Model IV) influenced fits to data in Fig. 3.
Except for k h , the best fit parameter values for Model IV agree well with the respective values obtained from Model I1 (Table 11). Model IV yields xf = 6 (compared to x? = 7 in Model 11; F = 5.2; 0.05 > p > 0.025) and a 6-fold larger value of kh; however, the large coefficient of variability for k h (Model 11) makes evaluation of the difference between k h values from Model I1 and Model IV difficult. The value of kc, the rate constant for conversion of (LR)s to (LR)s,c, was of the order of 1000 s" (with a >loo% coefficient of variability) indicating that the kc process is not rate limiting for internalization under conditions of experiment in Fig. 3.
On the basis of data in Table 11, we conclude that Model I (nonrecycling receptors) is not applicable to interactions of S. Vuk-Pavlovik, unpublished observation.

TABLE I1 Best jit parameters for interactions of human tumor necrosis factor-a with A549 cells
The best fit values were obtained by fitting equations of Models I-IV to data in Fig. 3. Values in italics denote fixed parameters. Other conditions as in Table I.
internal consistency of these best fit parameters is corroborated also by the fact that the calculated values of the ligandindependent ( [R] The numbers of data points/curve in Figs. 3 and 5 were sufficient for analysis by use of Durbin-Watson tables (15). The values of Durbin-Watson factor, d, for the best fit curves to data in Figs. 3 and 5 are given in Table IV. Comparison of values of d in Table IV with the respective values of dL and du shows that fits of Models I and I11 to data in Fig. 4 resulted in positive serial correlation of errors for curves fitted to surface-associated radioactivity; for the internalized radioactivity the test was inconclusive. This finding implies that Models I and I11 failed to account for the time dependence of surface-associated radioactivity in the interactions of T N F with A549 cells. For Models I1 and IV, the d values for internalized radioactivity were larger than d" implying negative serial correlation of errors; for surface-associated radioactivity, the test yielded inconclusive results. On the basis of Durbin-Watson analysis, Models I1 and IV were superior to Models I and I11 in describing the kinetics of T N F interactions with A549 cells. The results of Durbin-Watson analysis paralleled the xf values in Tables I1 and I11 because for data sets with smaller X:, no serially correlated errors in fits to internalized radioactivity were detected.

DISCUSSION
Studies leading to definition of elementary steps in interactions of selected protein ligands with cells have been extensively reviewed (cf. Refs. 10, 11, 16, 25). Dynamic aspects of these interactions have been considered ideal for application of mathematical modeling in a biological system (cf. Refs. 1,  10, 16, 25-27). In this paper we used the pre-steady state models of interactions of cytokines with cells to evaluate the contribution of receptor recycling to the maintenance of membrane receptor concentration. We measured the time-resolved concentration of radiolabeled ligands bound to membranereceptors and of internalized ligand-receptor complex. We restricted ourselves to this type of data because we intended to apply these and similar models to protein transport in three-dimensional tissue models; we expect that in such more complex experimental systems, time-dependent concentrations of not more than two compartments would be measured. We tested the applicability of models which took into account only experimentally verified compartments which could be individually measured by separation of the ligand bound to the membrane from internalized ligand. Thus, only the most essential phenomena in kinetics of interactions of cells with protein ligands were modeled; we neglected those which would have to be taken into account when convenient methods for more subtle time-resolved discrimination of cellular distribution of proteins become available. We believe that models presented in this paper are based on reasonable assumptions, are consistent with experimental evidence, are useful for derivation of relevant rate constants, and will serve as a basis for more refined and/or more realistic models. A quantitative comparison by kinetic modeling of interactions of biological response modifier proteins with human normal fibroblasts

TABLE 111
Best fit parameters for the interaction of human tumor necrosis factor-a with A549 cells The best fit values were obtained by simultaneous fitting of equations of Model I1 to two data sets (Fig. 5).  S, surface-associated radioactivity curve; I, internalized radioactivity curve; L and H refer to low concentration and high concentration data in Fig. 5, respectively. and epithelial tumor cells in vitro will be published separately.
Models I to IV are based on the assumption that each of the constituent elementary processes is continuous and adequately described by a first order rate constant. This assumption was validated experimentally for most rate constants considered in interactions of proteins with specific receptors, particularly for human epidermal growth factor (1, 11,16,26,28), transferrin and asialoorosomucoid (9, lo), and IFN (29).  (Tables 1-111). The x: value for TNF data (Model 11) was reduced significantly if data points for surface-associated radioactivity collected within first 15 min were disregarded (not shown); thus, it appears that early membrane-associated events are probably oversimplified in our models. We could not resolve the nature of this oversimplification as the introduction of a rate-limiting step which could function as a delay for internalization (Model IV) did not change the best fit parameter values or reduce the x? value (Table 11).
The best-fit curves to data in Figs. 3 and 5 were assessed by the Durbin-Watson analysis of serial correlation of residuals. It is noteworthy that curves of internalized radioactivity for Models I1 and IV displayed negative correlation of residuals showing that data were described by appropriate mathematical functions. Tests of the respective curves for surfaceassociated radioactivity yielded inconclusive results indicating more complex kinetics of the receptor-ligand interactions at the cell surface. However, refinement of kinetic equations must rely on advances in understanding of elementary steps of these interactions.
Our models include the influx of the ligand into the system and the efflux from it, but include also the influx and efflux of receptors (by the V, and kh processes, respectively). The influx and efflux of receptors were originally postulated by Wiley and Cunningham (1) in t,he model which was successfully applied to interactions of IFN with the nonrecycling Type I receptor in course of 6 h (6). The alternative assumption that the sum of receptors in all cellular compartments is constant was used in studies of recycling receptors for asialoglycoproteins and transferrin (9, lo), respectively. By use of experimentally derived rate constants, changes in concentration of total (occupied and free) surface receptors and free surface receptors in course o f 11 min of receptor cycling in absence of protein synthesis were successfully simulated (9, 10). The same assumption of constant total number of receptors was used by Gex-Fabry and DeLisi to formulate a more elaborate model of interactions of epidermal growth factor with fibroblasts (11,16). Decision on which of the models to apply to interactions of a particular ligand-cell pair will depend on experimental evidence for distinct patterns of receptor metabolism as well as on statisticaI criteria.
In conclusion, on the basis of time-resolved measurements of concentrations of protein ligands associated with the cell surface and internalized ligands we formulated simple mathematical models of the kinetics of ligand metabolism in the cell. By fitting equations of these mathematical models, we quantitated the contribution of receptor recycling to the maintenance of receptor concentrations on the surface of the cell, this method made it possible to discriminate between receptors which did recycle between the plasma membrane and interior of the cell and receptors which did not recycle. We studied how the best fit parameters depended on perturbation of the fixed input parameter; the k, and kr values were the most resilient, making it possible for the method to be applied particularly to studies of kinetic aspects of endocytosis and receptor recycling.