Bohr equation and the lost allosteric Bohr effects in symmetry

Abstract

Bohr, Hasselbalch and Krogh demonstrated a group of sigmoid curves under various carbon dioxide contents in 1904. Hill fitted these curves in 1910 with Hill equation without the physical meanings of Hill coefficient and dissociation constant. In 1965, Monod-Wyman-Changeux model (MWC) popularized the word “allostery” with 81 words of symmetry to define an orthosteric nature of cooperativity in a single and symmetric sigmoid curve. Paradoxically the MWC model didn’t quantify the homotropic Hill coefficient and confusingly described the symmetry of sigmoid shapes with three allosteric variables. A heterotropic Bohr equation, by clarifying the biophysical symmetry in allostery, suggests the solution of allosteric coefficients with only one Bohr variable. We reveal that the mathematical need of a fictional monomer by MWC model justify a symmetric logistic curve with a parabolic kernel of dissociation constant to model the 1904 sigmoid curves. The logistic-derived Bohr equation and its half-saturated P₅₀ equation successfully used the embedded P₅₀ values in the 1904 sigmoidal curves to quantify their hyperbolic conformational shifts and Hill coefficients (n) pending for a century. Both are the logarithmic functions of carbon dioxide. This truly quantitative Bohr equation digitizes the allosteric regulation of the orthosteric affinity by precisely cloning the original group of dissociation/association curves published in 1904. The Bohr equation honestly suggests that nature should have chosen the allosteric Bohr effects to modify hemoglobin to cope with the swift dynamic of gas exchange. The discovery of the Bohr function in Bohr equation challenges the feasibility of the orthosteric cooperativity of hemoglobin.