Commentary: The flexibility of SABRE, a new quantitative receptor function model, in fitting challenging concentration-effect data

In their recent paper (1), Olah and coworkers applied the SABRE model (2)(3)(4) to the analysis of their adenosine receptor response data measured previously (5) with three different agonists (NECA, CPA, and CHA) at seven concentrations ranging from 10 -10 to 10 -4 M before (N) and after (X) partial irreversible inactivation with FSCPX. Unequivocal fitting of this data obtained at different receptor levels (i.e., Furchgott's method) is particularly challenging because only a single inactivation level was used and one that resulted in no reduction of the maximal effect in any of the responses. After using an iterative approach involving four different fitting strategies, they concluded that "the SABRE model is at least as useful as two widely accepted older methods thought to have similar capabilities, the operational model of agonism and Furchgott's method, even if the quality of the data to be evaluated is somewhat challenging" (1).While the authors used a detailed and careful approach, the final SABRE fit obtained is not the best unified fit that can be achieved. As highlighted in their paper, "The first step in regression is to choose a proper model (equation)", and this necessarily involves choosing the right parameter setting. This is especially important with SABRE since it was intentionally designed to be a general model with multiple parameters that can and should be restricted for specific cases as needed: "its general form… can be reduced to consecutively nested simplified forms for special cases of its parameters…, and these can be used on their own when adequate" (3). For the present case, this would mean three parameters that are the same across all data, as they characterize the response (specifically, the Hill coefficient n, the pathway amplification γ, and the fraction of receptors inactivated q), and two that are the same for each agonists, as they characterize the agonists (specifically, the binding affinity constant Kd and the efficacy ε). This was done so only in their final fourth strategy; however, even there not as a single unified fit of the whole data as it involved first fitting "the data sets generated with the same agonist" to get the Kd estimates and then using these fixed "Kd values, provided by the third fitting strategy" to perform "a six-model global fitting" (1). Thus, it is not a single unified fitting as the Kd values are not fitted in the final step; they are retained at their constant values obtained in the previous fit of individual compound data. SABRE is not yet implemented in GraphPad Prism, the most widely used and powerful software for nonlinear regression for pharmacological-type data and used for these fittings; therefore, custom "user-defined equations" have to be used. Because Prism, in its current form, only allows parameters that are individually fitted for each set ("no constraint"), restricted to a common value across all data ("shared value for all data sets"), fixed as a single constant value ("constant equal to"), or fixed as constant for each set ("data set constant from column title"), its implementation for such complex data involving multiple agonists and receptor levels is not trivial. Thus, either separately defined equations have to be used for each set (column), as has been done here (see Supplementary Appendix in (1)), or a combination of custom ranges for one equation for each compound i (having the same Kd,i and the efficacy εi) with special column headers and corresponding conditional parameters for each inactivation j (to have the same qj) as has been done before to fit Furchgott-type data (e.g., Figure 4 in ( 4)).A single unified fitting of this dataset can be done by using either of these SABRE implementations as a more correct "fifth" strategy -this has been done here with the generous support of the authors who provided their original data and models. Results obtained with the original assumption that all three agonists are full agonists (i.e., all efficacies are equal to 1: εNECA= εCPA=εCHA=1) are presented in Table 1. While fit only improves slightly (as indicated by the decrease in the global sum of squared errors SSE from 20459 to 18354) and parameter values do not change significantly, this is a true unified fitting of all data, so that all parameters are obtained within a single fit. Undeniably, fit of these data remains challenging, and even with this unified SABRE fit ("approach 5"), parameters cannot be fully separated: dependency values are in the high or unacceptably high range (>0.9 and >0.99, respectively) with the sole exception of the fraction inactivated (q) value.With this implementation, even the three efficacies can be released and fitted allowing for the possibility of partial agonism; however, this results in only very minimal improvement in overall fit (SSE of 18142 vs 18354) and, due to the nature of data, in very uncertain parameter values; therefore, it is not included here. Nevertheless, it is worth mentioning that this fit indicated CPA and CHA as possibly less effective than NECA in producing this particular response. While all three are typically assumed to be full agonists, there are assays indicating possible functional selectivity and cases were "NECA was the most efficacious agonist… compared to the other agonists, although it had the lowest potency" (6). The difference in efficacies could also explain why the q value obtained for NECA via the classic Furchgott method is different from those of CPA and CHA (0.22 vs 0.11-0.13) or why the corresponding pharmacological shift ratios (Kd/EC50) are also 5-10 fold different (5).The main challenge with this dataset is that it does not allow adequate separation among efficacies, binding affinities, and amplification due to having only a single inactivation level and one that resulted in no reduction of the maximal effect in any of the responses. As noted by the authors (1) "For a reliable evaluation, the maximal effect after partial irreversible receptor inactivation is thought to have to be significantly smaller than the original maximal effect…". Nevertheless, SABRE is still Tables Table 1. Fit of the present data from (1) with SABRE assuming a single pathway (shared values for n=1, γ, and q) and full agonists (εNECA=εCPA=εCHA=1) each with their own Kd (Kd,NECA, Kd,CPA, and Kd,CHA).