A mathematical analysis of cooperativity and fractional saturation of oxygen in hemoglobin

Hemoglobin $(Hb)$ possesses good properties of cooperative system and it normally executes oxygen and other essential items via erythrocytes in the body. The chemical action of $Hb$ is to combine with oxygen (O2)(O2) in the lungs to form oxyhemoglobin (HbO2)(HbO2). Binding of oxygen with a hemoglobin is one of the important cooperative mechanism and is an emerging mathematical research area with wide range of applications in biomedical engineering and medical physiology. To this end, a mathematical model is proposed to study the fractional saturation of oxygen in hemoglobin to understand the binding effect and its stability at various stages. The mathematical formulation is based on the system of ordinary differential equations together with rate equations under different association and dissociation rate constants. The five states of the cooperative systems $Hb, HbO_2, Hb(O_2)_2, Hb(O_2)_3$ and $Hb(O_2)_4$ are modelled and the Hill’s function has been used to approximate the binding effect and saturation of ligand $(O_2)$ with respect to various rate constants. Also, the Adair equation has been employed to interpret the saturation concentrations of oxygen in hemoglobin.