A model-informed method to retrieve intrinsic from apparent cooperativity and project cellular target occupancy for ternary complex-forming compounds

Abstract

There is an increasing interest to develop therapeutics that modulate challenging or undruggable target proteins via a mechanism that involves ternary complexes. In general, such compounds can be characterized by their direct affinities to a chaperone and a target protein and by their degree of cooperativity in the formation of the ternary complex. As a trend, smaller compounds have a greater dependency on intrinsic cooperativity to their thermodynamic stability relative to direct target (or chaperone) binding. This highlights the need to consider intrinsic cooperativity of ternary complex-forming compounds early in lead optimization, especially as they provide more control over target selectivity (especially for isoforms) and more insight into the relationship between target occupancy and target response via estimation of ternary complex concentrations. This motivates the need to quantify the natural constant of intrinsic cooperativity (α) which is generally defined as the gain (or loss) in affinity of a compound to its target in pre-bound vs. unbound state. Intrinsic cooperativities can be retrieved via a mathematical binding model from EC₅₀ shifts of binary binding curves of the ternary complex-forming compound with either a target or chaperone relative to the same experiment but in the presence of the counter protein. In this manuscript, we present a mathematical modeling methodology that estimates the intrinsic cooperativity value from experimentally observed apparent cooperativities. This method requires only the two binary binding affinities and the protein concentrations of target and chaperone and is therefore suitable for use in early discovery therapeutic programs. This approach is then extended from biochemical assays to cellular assays (i.e., from a closed system to an open system) by accounting for differences in total ligand vs. free ligand concentrations in the calculations of ternary complex concentrations. Finally, this model is used to translate biochemical potency of ternary complex-forming compounds into expected cellular target occupancy, which could ultimately serve as a way for validation or de-validation of hypothesized biological mechanisms of action.

Fig. 1. Ternary complex-forming compounds can be grouped into multiple subtypes by their affinities to one or both proteins that constitute the ternary complex and the degree of induced positive or negative cooperativity. The proposed classification is idealized. In practice, ternary complex-forming compounds are often hybrids of two different types. In ternary complexes with similar thermodynamic stability, the molecular weight of the ternary complex-forming compound should inversely correlate with increasing cooperativity due to additional direct protein–protein or glue–protein interactions, which occur exclusively in the ternary complex. The colour gradient bars for “MW size” and “ADME quality” are intended to qualitatively illustrate the generally acknowledged trend that molecular weight and ADME quality are also inversely correlated (i.e., “smaller is better”).

Fig. 2. The proposed subtypes of ternary complex-forming compounds can be characterized by their main interactions and the corresponding degree of cooperativity: (a) bifunctionals are molecules with two binding functions of which each binds independently to either a chaperone or a target protein. Ternary complexes that bifunctionals form show no cooperativity (α = 1) as no direct target–chaperone, ligand–chaperone or ligand–target interactions are induced de novo. (b) Cooperative bifunctionals operate in a similar way as bifunctionals except that they exhibit cooperativity (α > 1). They induce direct target–chaperone or ligand–chaperone or ligand–target interactions potentially through more elaborate linkers (shorter, more rigid). The thermodynamic stability of ternary complexes formed by a cooperative bifunctional is higher than the stability resulting from the combination of individual compound affinities to chaperone and target. (c) Molecular glues type I only maintain a measurable intrinsic affinity to the chaperone but not to the target protein. However, ternary complexes from molecular glues type I often induce additional allosteric target chaperone interactions, i.e., protein–protein interactions in distance to the location of recruitment. The thermodynamic stability of the formed ternary complex mostly stems from a significant degree of cooperativity (α > 100). (d) Molecular glues type II hold neither affinities to the chaperone nor to the target protein and the two proteins show no intrinsic affinity to each other. The thermodynamic stability of the resulting ternary complex originates from interactions that occur exclusively in the ternary complex, i.e., due to its cooperativity, which is supposed to be high (α > 1000). (e) Molecular glues type III are, in contrast to the other subtypes, stabilizing an already existing, typically weak native intrinsic interaction between chaperone and target protein, which adds to the stability of the ternary complex. This type of glues has ideally no measurable intrinsic affinity to neither the chaperone nor the target protein. The mathematical approach to modelling ternary complexes formed by molecular glues of type III and hybrid forms between type I, II and III differ from the here presented and will be discussed elsewhere.

Fig. 3. The two main pathways that lead to the formation of the ternary complex CLT are passing through the formation of binary complexes CL and TL. The direct formation of CLT without the intermediate step of forming binary complexes is not considered here. Not discussed is a fourth case, in which the two proteins T and C form at first a binary protein complex TC then followed by a stabilizing ligand binding into CLT. The cooperativity α is defined as the ratio of the binary dissociation constants K_C,1 (or K_T,1) of ligand L and C (or T) to K_C,2 (or K_T,2) when ligand L is already prebound to T (as TL) or C (as CL).

Fig. 4. In closed systems like biochemical assays, the otherwise hard to determine free ligand concentration [L] is iteratively estimated by matching the given [L_tot] with [L_tot]_[L] = [L] + [CL]_[L] + [TL]_[L] + [CLT]_[L] that involves all individual species equations for CL, TL and CLT (top). When the ternary complex concentration is drawn with respect to free ligand concentrations, this curve is symmetric around the maximizing free ligand concentration [L]_max (bottom left). Depending on the chosen parameters, the corresponding total-to-free ligand transformation can show a high degree of non-linearity (bottom middle) resulting in a potential asymmetry of the ternary complex concentration curve with respect to total ligand (bottom right). Moreover, the total ligand concentration referring to the maximum of ternary complex [L_tot]_max is predicted to deviate from [L]_max. Compound parameters are referring to a standard cooperative bifunctional with K_T,1 = 10 000 nM, K_C,1 = 100 nM and α = 16; environmental parameters of [C_tot] = 25 000 nM and [T_tot] = 10 000 nM are representative of a biophysical assay.

Fig. 5. Left: A model-based correlation between concentrations of total and free ligand [L_tot] and [L] = [L^well_unbound] (purple line) shows strong deviation from the identity line (stippled line). Simulations were performed with the indicated values of experimental parameters [C_tot], [T_tot], K_C,1, K_T,1 and cooperativity α representative of a biochemical assay. The two values indicated on the plot (9.8 nM and 269 nM) are the calculated [L^well_unbound] when 100 or 1000 nM [L_tot] is applied, respectively. Right: The comparison of the same total ligand concentration [L_tot] in a biochemical and a cellular assay to assess the influence of the type of assay on the expected target occupancy. The two values indicated on the plot (8.2% and 33.5%) are the expected steady state target occupancy in a cellular assay when a [L_tot] = [L^well_unbound] = 9.8 or 100 nM is applied to the well, respectively.

Fig. 6. Biochemical and cellular assays differ in various aspects – most importantly, the biochemical assay represents a closed system whereas the cellular assay is considered an open system. In a biochemical assay that monitors ternary complex formation, the three species of chaperone C, target T and ligand L are homogenously dissolved and in equilibrium with each other. As ligand is incorporated into the binary (CL and TL) and ternary (CLT) complexes, the free ligand concentration in the well [L^well_unbound] can become significantly lower than the initially applied [L_tot], depending upon the assay conditions. In contrast, a cellular assay is an open system due to the exchange between the cell and the media in the well. As free ligand can permeate through the membrane into the cell, [L^cell_unbound] increases until it corresponds to [L^well_unbound]. For well volumes that are exceedingly larger than the total cell volume, the monitorable [L_tot] corresponds under equilibrium conditions to [L^well_unbound] and the latter to [L^cell_unbound].

Fig. 7. Simulations of bound chaperone without counter protein present ([T_tot] = 0 nM). Results are shown for percentage of total bound chaperone (left) and for each chaperone-containing complex (CL and CLT, right). As this is a binary binding experiment, total bound chaperone is solely due to formation of the CL complex ([CLT] = 0).

Fig. 8. Simulations of bound chaperone, as in Fig. 7, but with counter protein present at a 200-fold excess over chaperone ([T_tot] = 1000 nM) and a cooperativity of α = 2. Results are shown for percentage of total bound chaperone (left) and for each chaperone-containing complex (CLT and CL, right).

Fig. 9. Simulations of bound chaperone, as in Fig. 7, but now configured as a non-cooperative system (α = 1) and an equivalent affinity for each binary complex (K_C,1 = K_T,1 = 100 nM). Results are shown for percentage of total bound chaperone (left) and for each chaperone-containing complex (CLT and CL, right). No EC₅₀ shift is observed as because TL binds with the same affinity to C as to L alone (left). Therefore, the %C_bound (or the %T_bound) curve remains unchanged with an EC₅₀ at 100 nM (left) despite significant CLT formation (right).

Fig. 10. Simulations of bound target without counter protein present. Results are shown for percentage of total bound target (left) and for each target-containing complex (right). As this is a binary binding experiment, total bound target is solely due to formation of the TL complex.

Fig. 11. Simulations of bound target, as in Fig. 10, but now configured with an excess of the chaperone counter protein. Results are shown for percentage of total bound target (left) and for each target-containing complex (CLT and TL, right).