Journal of Theoretical Biology 229 (2004) 127–138 Analysis of compartmental models of ligand-induced endocytosis Abraham R. Tzafriri a, *, David Wu a , Elazer R. Edelman a,b a Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Room 16-343, Cambridge, MA, USA b Cardiovascular Division, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA Received 24 October 2003; received in revised form 4 March 2004; accepted 12 March 2004 Abstract Kinetic models have played a pivotal role in the study of ligand-induced endocytosis. However, an analysis that suggests a systematic way to validate such models is lacking. The current work analyses the base model of ligand-induced endocytosis for three widely used experimental protocols. In protocol I cells initially devoid of ligand are incubated in ligand solution, whereas protocols II and III are desorption experiments in which an initial pool of surface or internalized ligand–receptor complexes, respectively, are released into an elution medium that is initially devoid of ligand. A short-time analysis of protocol I using successive substitutions yielded a corrected pre-factor for the In/Sur plot introduced by Wiley and Cunningham (Cell 25 (1981) 433). In contrast, neglecting the variation in receptor numbers yielded an approximation of protocol I that is valid for long times (e.g. tens of minutes). Similarly, the low cell-concentration limits of protocols II and III are derived by neglecting the concentration of free ligand. The simplicity of these approximations provides a simple and reliable method for estimating the parameters governing ligand kinetics, while their definitive nature implies that they can be used to verify the validity of the base model. This analysis also provides insight on the fast endocytosis and recycling limit of protocol III. r 2004 Elsevier Ltd. All rights reserved. Keywords: Analytical approximations; Initial transient; Linear stability; In/Sur plot; Growth factors 1. Introduction Most models of cellular pharmacokinetics are based on an analogy between receptor and enzyme kinetics and employ kinetic equations to describe a small number of rate limiting steps between various cellular compartments (Levitzki, 1984; Lauffenburger and Linderman, 1993; Mukherjee et al., 1997; Wiley et al., 2003). In addition to numerical simulations of these models, their relative simplicity has prompted the derivation of simple analytical approximations for purposes of parameter estimation and model validation (Wiley and Cunningham, 1981,1982; Zigmond et al., 1982; Shimizu and Kawashima, 1989). Motivated by the analogy to enzyme kinetics, Wiley and Cunningham (1981) employed the quasi-steady-state approximation to derive a graphical method for parameter estimation. These authors also introduced the In/Sur plot as a means of estimating the endocytosis rate constant from simple ligand sorption experiments (Wiley and Cunning- ham, 1982). A major limitation of the In/Sur plot is that it is only valid for short times such that degradation and recycling terms are negligible compared to the endocy- tosis and binding terms. As the number of parameters in receptor signaling models increases, numerical techniques for parameter estimation are increasingly employed (Waters et al., 1990; Schoeberl et al., 2002; Swameye et al., 2003). However, this approach has two limitations. First, just as in enzyme kinetics, the stiffness of these models implies that only a subset of the parameters can be estimated simultaneously for a given experiment. This is because by its very definition stiffness implies the existence of a hierarchy of time scales (Heinrich et al., 1977; Schauer and Heinrich, 1979; Roussel and Fraser, 1991). Consequently, most experiments only capture the slow quasi-steady-state kinetics of stiff systems. In the absence of a good characterization of the kinetic equations it is impossible to plan a logical experimental strategy for parameter fitting and a brute force numerical approach may require many experimental permutations. Second, a fully numerical approach of ARTICLE IN PRESS *Corresponding author. Tel.: +1-617-252-1655; fax: +1-617-253- 2514. E-mail address: ramitz@mit.edu (A.R. Tzafriri). 0022-5193/$-see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.03.009