Modelling negative cooperativity of IGF-1R and IR

The experimental data for negative cooperativity of IGF-1R and IR were those described previously¹⁸ and are provided for completeness here in Supplementary Table ⁴ by permission of Professor Pierre De Meyts. As indicated in the Discussion, the initial binding of ligand to the “closed” form of receptor can be explained by an induced fit model or by transient receptor opening, the latter effectively being described by the HO model¹⁸ upon reversal of the percentage of times that the receptor spends in its respective open and closed conformations. The exact nature of this binding event is not important for the modelling presented here, as its sequential components can be grouped into a single reaction (Supplementary Figure ^5a) that represents high-affinity receptor binding (and receptor activation). Binding of a second ligand would then lead to either an asymmetric or symmetric receptor conformation. Both cases need to be considered. (i) In case of an asymmetric conformation, the second ligand hypothetically binds to a partially open site 1 of the alternate pair of binding sites, without it engaging site 2 (Supplementary Figure ^5b). This interaction is expected to have a lower affinity to that of an interaction engaging both sites. In order for the negative cooperativity to occur, the asymmetric conformation is presumed to transition between the two possible states in which the ligand initially bound with high affinity disengages site 2 (leading to low affinity) and the ligand initially bound with low affinity to site 1 alone engages both sites (leading to high affinity) (Supplementary Figure ^5b). This mechanism is formally identical to negative cooperativity within HO model, and thus the HO formalism can be applied (albeit with an alternative structural interpretation). (ii) In the case of a symmetric conformation (Supplementary Figure ^5c), the two sites have ligand bound with identical affinity. This affinity is expected to be reduced compared to that of the singly bound receptor, as otherwise we would have a receptor with two high-affinity sites, contradicting binding data that demonstrate that there is only one high-affinity site per holo-receptor^18,38,56, unlike the soluble IR ectodomain that has two equal lower-affinity sites⁵⁷. Indeed, it is plausible that symmetrical opening of the receptor domains to accommodate two ligands requires distortion of the receptor structure in energetically costly fashion that reduces ligand affinity. The binding of a third insulin molecule is proposed to account for the ascending phase of accelerated dissociation for IR³⁸. IGF-1R lacks this part of the curve and thus, for simplicity, binding of the third ligand will be considered only in the case of insulin binding to IR. Additional separation (“opening”) of the receptor domains may be required to accommodate the third ligand (Supplementary Figure ^5d), presumably via an energetically unfavourable process that results in very low affinity for that ligand. It is proposed that binding of the third ligand “locks” the tracer in the bound state in the experiment for accelerated dissociation, and tracer dissociation can only occur after the cold ligand dissociates³⁸. Taking into account the above described binding reactions, the model proposed here with the use of doubly liganded, symmetrical receptor conformation leads to a compact binding scheme of the ligand–receptor interaction (Supplementary Figure ^5e). It should be noted that this binding scheme is applicable only to the experimental conditions described above. For example, receptor intermediaries with two or three hot ligand molecules bound were excluded from the reaction scheme, since they would not be formed in any significant quantities at 10 pM ligand concentration. Similarly, intermediates with only cold ligand molecules bound were eliminated due to tracer pre-binding. Endocytosis is, however, included, as even though the binding data were derived from experiments performed at 16 °C, endocytosis at this temperature cannot be totally excluded¹⁸. Thus, as within the HO model, it is assumed that upon activation of inactive receptor intermediary, R₀₀₀, the active intermediaries such as R_h00, R_0c0, R_hc0 or R_hcc (see Supplementary Figure ^5e) are internalized with an internalization rate constant k_end. Upon internalization, it is assumed that ligand dissociates instantly which leads to accumulation of hot ligand, Lig_end, and internalized receptor, R_cyt, inside the cells. The internalized receptor, R_cyt, is recycled back to the plasma membrane with an exocytosis rate constant, k_ex. The internalized ligand, Lig_end, is recycled out of cells (either intact or degraded) with an exocytosis rate constant, k_ex. The binding of two species of ligand (hot and cold) in the presence of endo- and exocytosis and under conditions of no ligand depletion can be described by a system of ordinary differential equations shown in Supplementary Figure ⁶. The rate constants for endocytosis and exocytosis in IM9 cells were taken from the HO model¹⁸. The initial values for a₁ and d₁ (high affinity) site, a₂ and d₂ (low affinity symmetrical conformation) and a₃ and d₃ (describing binding of the third insulin molecule) were also taken from the HO model¹⁸ and manually optimized to achieve a fit to experimental data for accelerated dissociation at 20 min while keeping the high-affinity site constrained to K_d = 0.12 nM for IGF-I and K_d = 0.2 nM for insulin and the low-affinity site to K_d = 4.3 nM in case of IGF-I and K_d = 6 nM in case of insulin. Simulations were performed using Mathematica v11.0 (Wolfram). The optimized parameter values are shown in Supplementary Table ⁵. No attempt was made to obtain a best fit of parameters or to establish if the identified parameter set is unique; nevertheless, the identified set of parameters leads to good agreement with experimental data (Fig. 8b).