The insulin and insulin-like growth factor 1 receptors activate overlapping signalling pathways that are critical for growth, metabolism, survival and longevity. Their mechanism of ligand binding and activation displays complex allosteric properties, which no mathematical model has been able to account for. Modelling these receptors' binding and activation in terms of interactions between the molecular components is problematical due to many unknown biochemical and structural details. Moreover, substantial combinatorial complexity originating from multivalent ligand binding further complicates the problem. On the basis of the available structural and biochemical information, we develop a physically plausible model of the receptor binding and activation, which is based on the concept of a harmonic oscillator. Modelling a network of interactions among all possible receptor intermediaries arising in the context of the model (35, for the insulin receptor) accurately reproduces for the first time all the kinetic properties of the receptor, and provides unique and robust estimates of the kinetic parameters. The harmonic oscillator model may be adaptable for many other dimeric/dimerizing receptor tyrosine kinases, cytokine receptors and G-proteincoupled receptors where ligand crosslinking occurs.

Current evidence supports a binding model in which the insulin molecule contains two binding surfaces, site 1 and site 2, which contact the two halves of the insulin receptor. The interaction of these two surfaces with the insulin receptor results in a high affinity cross-linking of the two receptor ␣ subunits and leads to receptor activation. Evidence suggests that insulin-like growth factor-I (IGF-I) may activate the IGF-I receptor in a similar mode. So far IGF-I residues structurally corresponding to the residues of the insulin site 1 together with residues in the C-domain of IGF-I have been found to be important for binding of IGF-I to the IGF-I receptor (e.g. results in a significant reduction in IGF-I receptor binding affinity, whereas alanine substitution at position 53 had no effect on IGF-I receptor binding. The multiple substitutions resulted in a 33-100-fold reduction in IGF-I receptor binding affinity. These data suggest that IGF-I, in addition to the C-domain, uses surfaces similar to those of insulin in contacting its cognate receptor, although the relative contribution of the side chains of homologous residues varies. IGF-I2 is a single chain polypeptide that plays an important role in regulating growth and development (1, 2). IGF-I consists of 70 amino acid residues arranged in four domains. The A and B domains are homologous to the A and B chains of insulin (Fig. 1), whereas the C-domain connecting the A-and B-domain and the D-domain extending from the C-terminal end of the A-domain are only present in IGF-I (3-5). The three-dimensional structures of insulin and IGF-I are also very similar. The B-domain of the two molecules is arranged in a central ␣-helix including residues 8 -17 (B9 -B19) (insulin residues in parentheses), and the A-domain contains two anti-parallel ␣-helices including residues 43-48 (A1-A8) and residues 54 -60 (A13-A20) (4, 6, 7).The insulin receptor family consists of the insulin receptor (IR), insulin-like growth factor-I receptor (IGF-IR) and the insulin-receptor-related receptor, all of which are receptortyrosine kinases. The members of the insulin receptor family consist of two receptor halves, each comprising an extracellular ␣-subunit and a transmembrane ␤-subunit, linked by a disulfide bridge. The receptor exists as a dimer when no ligand is bound, making an ␣2␤2 receptor held together by disulfide bridges between the two ␣-subunits. The IR and the IGF-IR have sequence similarities varying from 41 to 84% depending on which regions are being compared. The overall structure of the L1-Cys-rich-L2 domains of the two receptors have been shown by x-ray crystallography to be similar (8 -11).Numerous studies indicate that the IR and IGF-IR contains two separate binding sites for the ligand; however, only one ligand binds and activates the ␣2␤2 receptor dimer. Insulin is believed to interact with the receptor using two binding surfaces, which cross-link the two receptor halves. This cross-

Insulin and the insulin-like growth factors (IGFs) bind with high affinity to their cognate receptor and with lower affinity to the noncognate receptor. The major structural difference between insulin and the IGFs is that the IGFs are single chain polypeptides containing A-, B-, C-, and D-domains, whereas the insulin molecule contains separate A-and B-chains. The C-domain of IGF-I is critical for high affinity binding to the insulin-like growth factor I receptor, and lack of a C-domain largely explains the low affinity of insulin for the insulin-like growth factor I receptor. It is less clear why the IGFs have lower affinity for the insulin receptor. In this study, 24 insulin analogues and four IGF analogues were expressed and analyzed to explore the role of amino acid differences in the Aand B-domains between insulin and the IGFs in binding affinity for the insulin receptor. showed affinities similar to that of insulin for the insulin receptor. The mitogenic potency of these analogues correlated well with the binding properties. Thus, a small number of A-and B-domain substitutions that map to the IGF surface equivalent to the classical binding surface of insulin weaken two hotspots that bind to the insulin receptor site 1.

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Heterozygosity for this novel IGF1 mutation in children born from a mother with the same mutation, presumably in combination with other genetic factors for short stature, leads to severe short stature, which can be successfully treated with GH.

The insulin/relaxin superfamily of peptide hormones comprises 10 members in humans. The three members of the insulin-related subgroup bind to receptor tyrosine kinases (RTKs), while four of the seven members of the relaxin-like subgroup are now known to bind to G-protein-coupled receptors (GPCRs), the so-called relaxin family peptide receptors (RXFPs). Both systems have a long evolutionary history and play a critical role in fundamental biological processes, such as metabolism, growth, survival and longevity, and reproduction. The structural biology and ligand-binding kinetics of the insulin and insulin-like growth factor I receptors have been studied in great detail, culminating in the recent crystal structure of the insulin receptor extracellular domain. Some of the fundamental properties of these receptors, including constitutive dimerization and negative cooperativity, have recently been shown to extend to other RTKs and GPCRs, including RXFPs, confirming kinetic observations made over 30 years ago.

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