Using the MWC model to describe heterotropic interactions in hemoglobin

Quantitative Model-Based Assessment of Multiple Sickle Cell Disease Therapeutic Approaches Alone and in Combination

Three sickle cell disease (SCD) treatment strategies, stabilizing oxygenated hemoglobin (oxyHb), lowering 2,3-BPG, and inducing fetal hemoglobin (HbF) expression aim to prevent red blood cell (RBC) sickling by reducing tense-state sickle hemoglobin that contributes to polymer formation. Induction of 30% HbF is seen as the gold standard because 30% endogenous expression is associated with a lack of symptoms. However, the level of intervention required to achieve equivalent polymerization protection by the other strategies is uncertain, and there is little understanding of how these approaches could work in combination. We sought to develop an oxygen saturation model that could assess polymerization protection of all three approaches alone or in combination by extending the Monod-Wymann-Changeux model to include additional mechanisms. Applying the model to monotherapies suggests 51% sickle hemoglobin (HbS) occupancy with an oxyHb stabilizer or lowering RBC 2,3 BPG concentrations to 1.8 mM would produce comparable polymerization protection as 30% HbF. The model predictions are consistent with observed clinical response to the oxyHb stabilizer voxelotor and the 2,3-BPG reducer etavopivat. The model also suggests combination therapy will have added benefit in the case of dose limitations, as is the case for voxelotor, which the model predicts could be combined with 20% HbF or 2,3-BPG reduction to 3.75 mM to reach equivalent protection as 30% HbF. The proposed model represents a unified framework that is useful in supporting decisions in preclinical and early clinical development and capable of evolving with clinical experience to gain new and increasingly confident insights into treatment strategies for SCD.

Mathematical models describing oxygen binding by hemoglobin

Despite the fact that the investigation of the structural and functional properties of hemoglobin dates back more than 150 years, the topic has not lost its relevance today. The most important component of these studies is the development of mathematical models that formalize and generalize the mechanisms determining the cooperative binding of ligands based on data on the structural and functional state of the protein. In this work, we review the mathematical relationships describing oxygen binding by hemoglobin, ranging from the classical Hüfner, Hill, and Adair equations to the Szabo-Karplus and tertiary two-state mathematical models based on the Monod-Wyman-Changeux and Koshland-Némethy-Filmer concepts. The generality of the considered equations as mathematical functions, bearing in their basis a power dependence, is demonstrated. The problems and possible solutions related to approximation of experimental data by the oxygenation equations with correlated fitting parameters are noted. Attention is paid to empirical equations, extended versions of the Hill equation, where the coefficient of cooperation is modulated by Gauss and Lorentz distributions as functions of partial oxygen pressure.

Partitioning the Hill coefficient into contributions from ligand-promoted conformational changes and subunit heterogeneity

Heterooligomers that undergo ligand-promoted conformational changes are ubiquitous in nature and involved in many essential processes. Conformational switching often leads to positive cooperativity in ligand binding that is reflected in a Hill coefficient with a value greater than one. The subunits comprising heterooligomers can differ, however, in their affinity for the ligand. Such so-called site heterogeneity results in apparent negative cooperativity that is reflected by a Hill coefficient with a value less than one. Consequently, positive cooperativity due to the ligand-promoted allosteric switch can be masked, in cases of such heterooligomers, by apparent negative cooperativity owing to site heterogeneity. Here, we derived expressions for the Hill coefficient, in the case of a heterodimer, in which the contributions from the ligand-promoted allosteric switch and site heterogeneity are separated. Using these equations and simulations for higher order oligomers, we show under which conditions site heterogeneity can significantly mask the extent of observed positive cooperativity.

Quantitative dissection of transcription in development yields evidence for transcription-factor-driven chromatin accessibility

Thermodynamic models of gene regulation can predict transcriptional regulation in bacteria, but in eukaryotes, chromatin accessibility and energy expenditure may call for a different framework. Here, we systematically tested the predictive power of models of DNA accessibility based on the Monod-Wyman-Changeux (MWC) model of allostery, which posits that chromatin fluctuates between accessible and inaccessible states. We dissected the regulatory dynamics of hunchback by the activator Bicoid and the pioneer-like transcription factor Zelda in living Drosophila embryos and showed that no thermodynamic or non-equilibrium MWC model can recapitulate hunchback transcription. Therefore, we explored a model where DNA accessibility is not the result of thermal fluctuations but is catalyzed by Bicoid and Zelda, possibly through histone acetylation, and found that this model can predict hunchback dynamics. Thus, our theory-experiment dialogue uncovered potential molecular mechanisms of transcriptional regulatory dynamics, a key step toward reaching a predictive understanding of developmental decision-making.

Evolutionary and functional insights into the mechanism underlying body-size-related adaptation of mammalian hemoglobin

Hemoglobin (Hb) represents a model protein to study molecular adaptation in vertebrates. Although both affinity and cooperativity of oxygen binding to Hb affect tissue oxygen delivery, only the former was thought to determine molecular adaptations of Hb. Here, we suggest that Hb affinity and cooperativity reflect evolutionary and physiological adaptions that optimized tissue oxygen delivery. To test this hypothesis, we derived the relationship between the Hill coefficient and the relative affinity and conformational changes parameters of the Monod-Wymann-Changeux allosteric model and graphed the 'biophysical Hill landscape' describing this relation. We found that mammalian Hb cooperativity values all reside on a ridge of maximum cooperativity along this landscape that allows for both gross- and fine-tuning of tissue oxygen unloading to meet the distinct metabolic requirements of mammalian tissues for oxygen. Our findings reveal the mechanism underlying body size-related adaptation of mammalian Hb. The generality and implications of our findings are discussed.

GroEL Allostery Illuminated by a Relationship between the Hill Coefficient and the MWC Model

A fundamental problem that has hindered the use of the classic Monod-Wyman-Changuex (MWC) allosteric model since its introduction is that it has been difficult to determine the values of its parameters in a reliable manner because they are correlated with each other and sensitive to the data-fitting method. Consequently, experimental data are often fitted to the Hill equation, which provides a measure of cooperativity but no insights into its origin. In this work, we derived a general relationship between the value of the Hill coefficient and the parameters of the MWC model. It is shown that this relationship can be used to select the best estimate of the true combination of the MWC parameter values from all the possible ones found to fit the data. Here, this approach was applied to fits to the MWC model of curves of the fraction of GroEL molecules in the high-affinity (R) state for ATP as a function of ATP concentration. Such curves were collected at different temperatures, thereby providing insight into the hydrophobic effect associated with the ATP-promoted allosteric switch of GroEL. More generally, the relationship derived here should facilitate future thermodynamic analysis of other MWC-type allosteric systems.