(Solved): Hemoglobin oxygen binding. Hemoglobin is a symmetrical tetrameric protein with 4 oxygen binding ...

Hemoglobin oxygen binding. Hemoglobin is a symmetrical tetrameric protein with 4 oxygen binding sites (see figure 1) that exhibits a highly cooperative binding curve. Figure 1: Schematic figure of a hemoglobin protein, with the oxygen binding sites (called heme) in red and binding oxygens in light blue When one oxygen binds to one of these four sites, a local deformation occurs that lowers the energy of the second binding site. Taking this into account, we can make a model with the following assumptions: - A hemoglobin molecule with no oxygen bound has an energy of zero. - A hemoglobin molecule has an energy $?<0$ if only one oxygen is bound to any of the four binding sites. - When any two sites are occupied the energy is now $J<2?<0$. - Similarly, when any three sites are occupied, the energy is $K<\frac{3}{2}J<3?<0$, and is $L<\frac{4}{3}K<2J<4?<0$ when all four sites are occupied. Suppose we have an ensemble of $N$ hemoglobin proteins: (a) [4] Write an expression for the grand canonical partition function for this model.