Up- and Down-regulation of Insulin Receptors KINETIC MODELS*

A theoretical model for insulin receptor synthesis and degradation in differentiating 3T3-Ll adipocytes is described. This three-step irreversible ordered se- quence model explains the up- and down-regulation of receptors in terms of the level of insulin concentration. Kinetic expressions were derived for the model. Numerical solutions for these equations, based on data reported by Reed and Lane (Reed, B. C., and Lane, M. D. (1980) Proc. Natl. Acad. Sci. U. S. A. 77,285-289) were used for computer-generated curves illustrating insulin-dependent receptor synthesis and degradation. Results show that this model provides the best fit to the reported data and lend support to the suggestion that the free recycled receptor may differ from the newly synthesized receptor. A possible role for the recycled receptor in signal modulation is suggested.

The chronic exposure of cells to insulin in vitro causes a decrease, or down-regulation (3,17,25,26) in the level of cell surface insulin receptors. This correlation between ambient insulin concentration and surface insulin receptor number extends to other cell types studied in culture or in uitro (3,(17)(18)(19)25,26). Kosmokos and Roth (27) and Van Obberghen et al. (28)  tion may be responsible for such down-regulation. Evidence such as recovery from down-regulation upon the removal of insulin (3,15), differentiation-induced increases in surface receptor levels of thymocytes (29) and 3T3-Ll preadipocytes (18,19,(30)(31)(32), and drug-induced increases (33,34) suggests that cells also have the capacity to up-regulate the number of insulin receptors. Ronnett et al. (19) have shown that the rate constants for inactivation of the cell surface and total cellular insulin receptors were identical in the up-regulated state or in the down-regulated state and that the rate-limiting step in the receptor inactivation pathway occurs at the cell surface.
The heavy isotope "density shift" technique (35-37) has been used to assess the effect on the rate of synthesis and degradation of the insulin receptor (14)(15)(16)(17)(18)(19). It has been suggested that insulin-induced receptor down-regulation involves the translocation of insulin receptors from the cell surface to an intracellular location (18,19).
In this communication, we describe kinetic models for the regulation of insulin receptors that include the redistribution of cell surface receptors to intracellular compartments and modulation of receptor synthesis, and we examine the possible ramifications of receptor recycling in a number of cellular processes.

Two-step Irreversible Ordered Sequence Model (Reed and Lane) (14)"In heavy
isotope labeling studies of receptor synthesis, Reed and Lane (14) have proposed a kinetic model ( Assuming that the kinetic model is a three-step irreversible sequence in which k-* << k2, rate expressions were expressed and solved for stated conditions, as shown in the Appendix (Equations 1-10).' The insulin dosage-dependent regulation of cell receptors as a function of time may be expressed as Total concentration of receptor may be evaluated from Using these equations (derived in the Appendix), it is possible to predict the insulin-dependent kinetics of heavy (new) receptor synthesis and light (old) receptor degradation in cultured cells based on the experimental data of Reed and Lane (14) collected by the heavy isotope density-shift method.
Kinetic Model for Receptor Regulation-Certain conditions must be imposed in order to maintain a steady state level of free receptor concentration. Case (i), a three-step irreversible Two methods were used to solve the differential equations (38). In the first, the equations were simplified by the as- Using Equation 1 (14), the receptor concentration versus time was also plotted and compared with that obtained from the proposed system.
In the second method, the three differential equations were solved simultaneously using the fourth order Runge-Kutta formula (39). Here At, the step-size time increment, was set at 0.05 h and the error obtained using At/2 was found to be 1. The insulin-dependent kinetics of heavy and light receptor synthesis and degradation are shown in Fig. 1 of the Appendix. As the insulin concentration is decreased, both the dipping of these curves and the time lag period become more pronounced. This figure, therefore, represents the kinetic model described in case (i) and is based on Equations 2 and 3.
The up-and down-regulation kinetics of receptor synthesis is shown in Fig. 3 state (see also Appendix Fig. 2). Under these conditions, it is evident that the recycling of receptor is minimal, so that both insulin and receptor undergo continued degradation. The kinetic model of case (iii), in which the total free receptor concentration is maintained at a high level, is best described by Fig. 3 of the Appendix, in which the free receptor recycling appears to modulate the receptor lifetime and insulin transport through the membrane. This model provides the best fit to the Lane and Reed data and has the added advantage of describing up-and down-regulation without changing kd, the first order rate constant for the degradation of insulin receptor.

DISCUSSION
The three-step irreversible ordered sequence model for the regulation of insulin receptor explains the up-and downregulation of receptor in terms of insulin concentration, thus eliminating some of the difficulties inherent in the two-step model (14).
In order to maintain the level of free insulin receptor in a steady state, this kinetic model describes a reciprocal relationship between the total number of receptors and the insulin concentration, a relationship known as ligand-induced downregulation of insulin binding, which may be formulated as light receptor synthesis and degradation, but in this case the The free receptor generated from recycling the RZ complex kinetic plots were generated by the numerical solution of the may differ from the receptor newly synthesized from amino two differential equations for [RZ],. Here the insulin dependacids. It is possible, therefore, that the recycled receptor may ence of light receptor degradation, [RZ]l = [RZIT - [RZ],, is have the option of binding or not binding to insulin, thus evident as in the kinetic model described in case (ii). The free signaling transport activation and modulating receptor synreceptor concentration is no longer maintained at a steady thesis. Such a conclusion would be consistent with the insu-

Insulin Receptor Regulation
lin-dependent degradation of light and heavy receptor shown in Fig. 4 (and Fig. 2 of the Appendix).
While a theoretical kinetic model (see Appendix Fig. 4) such as this suggests the possibility that the recycled receptor may play a role in the modulation of further transport activation, it remains to be determined in further studies whether there is any difference in the biological activity of light and heavy receptors which would provide experimental evidence to support such a conclusion.