Using the MWC model to describe heterotropic interactions in hemoglobin

Abstract

Hemoglobin is a classical model allosteric protein. Research on hemoglobin parallels the development of key cooperativity and allostery concepts, such as the 'all-or-none' Hill formalism, the stepwise Adair binding formulation and the concerted Monod-Wymann-Changuex (MWC) allosteric model. While it is clear that the MWC model adequately describes the cooperative binding of oxygen to hemoglobin, rationalizing the effects of H+, CO2 or organophosphate ligands on hemoglobin-oxygen saturation using the same model remains controversial. According to the MWC model, allosteric ligands exert their effect on protein function by modulating the quaternary conformational transition of the protein. However, data fitting analysis of hemoglobin oxygen saturation curves in the presence or absence of inhibitory ligands persistently revealed effects on both relative oxygen affinity (c) and conformational changes (L), elementary MWC parameters. The recent realization that data fitting analysis using the traditional MWC model equation may not provide reliable estimates for L and c thus calls for a re-examination of previous data using alternative fitting strategies. In the current manuscript, we present two simple strategies for obtaining reliable estimates for MWC mechanistic parameters of hemoglobin steady-state saturation curves in cases of both evolutionary and physiological variations. Our results suggest that the simple MWC model provides a reasonable description that can also account for heterotropic interactions in hemoglobin. The results, moreover, offer a general roadmap for successful data fitting analysis using the MWC model.

Fig 1. Successful application of the three-equation system (TES) strategy in the case of evolutionary variations in hemoglobin.

(A) Correlation plot relating the values of n_H for the 13 different mammalian hemoglobins of the physiologically-sound solution set derived either using the Hill equation (as reported in ref. [33]) or calculated according to the MWC model, based on L, K_R and K_T values (Table 1, red columns). Values for the human and elephant species correspond to the averaged values of three independent triplicates. (B) The same analysis but for the non-physiological solution set (Table 1, rightmost black columns). The expression for n_H in terms of the L, K_R and K_T model parameters is known and can be obtained using the $\partial ({\overline{Y}}_{MWC}/1-{\overline{Y}}_{MWC})/\partial \left(\text{log}\left[S\right]\right)$ Hill transformation (see Eq 4 and Methods).

Fig 2. The three-equation system strategy is inadequate for assessing the effect(s) of physiological variations on allosteric protein function.

(A-B) Correlation plot relating the observed (open circles) and calculated (filled circles) n_H values of the different dataset oxygenation curves to either pH (A) or 2,3-BPG (B) effector concentrations (S1 Table; see Methods). (C-D) Dependence of the L and c parameters of the physiological datasets on pH (C) and 2,3-BPG concentrations (D). A similar analysis of additional pH and 2,3-BPG physiological datasets, collected under different experimental conditions (see S1 Table), is presented in S2 Fig.

Fig 3. Successful application of global fitting analysis of hemoglobin physiological datasets.

(A) Global fitting analysis of the human hemoglobin pH, CO₂ and 2,3-BPG physiological datasets using the traditional form of the MWC equation. Source data is indicated above each panel. (B) Dependence of observed (open circles) and calculated (filled circles) Hill values of each dataset on effector concentration. (C) Dependence of the apparent L values of each physiological dataset on effector concentration. Solid curves represent the results of curve fitting to the MWC-derived equation $L_{app}=L_o{((1+\left[I\right]/K_I^T)/(1+\left[I\right]/K_I^R))}^4$, assuming non-exclusive binding of the inhibitor effector (I) to both the T and R MWC conformations [11,12]. In the case of organophosphate inhibitors, a power of one was used in the above equation, as only one site is available to BPG for binding to hemoglobin.