Question

Dimoglobin as a case study in cooperativity.

Note that each binding site can be occupied, $$\sigma_i$ = 1$ , or unoccupied, $0;=0$ , and that a parameter J describes the cooperativity between the O2 molecules when both states are occupied (i.e., the energy is not just the sum of the individual binding energies).

a) Write down the weights (Gibbs factors) for each of the different states for the dimoglobin system shown in Figure 2.

b) What is the formula that describes the energy of the dimoglobin system?

c) Obtain the grand partition function for the dimoglobin system.

Answer 1

Cooperativity is a phenomenon of interaction process by which binding of a ligand to one site of a macromolecule (enzyme, receptors, etc) influences binding at a second site. When a ligand bings to a protein it will induce some conformational changes, as a result, when the next ligand binds, it finds an altered protein interface and hence experiences a different binding energy.The binding affinity of oxygen for haemoglobin depends upon whether or not other oxygens are already bound. In the interaction of ligand receptor binding in haemoglobin there is a need of two state occupation variables, where $\sigma _{i}$ takes the value of 0 (unbound) or 1 (bound) characterising the occupancy of site i within the molecule.

The dimeric haemoglobin (dimoglobin) has two O2 binding sites. This molecule is characterized by four distinct states to each of the binding sites of the dimoglobin molecule being either occupied or empty according to the given toy model of dimoglobin.

(a) For state 1, the weight or Gibbs factor is 1, as the both site is unoccupied. For state 2 and 3, where one site is occupied by O2, the weights or Gibbs factor is
e-Ble-H)
. For state 3, where both the binding sites are occupied by O2, the weights or Gibbs factor is
e-3(24+-21)
. Each state of occupancy is characterized by a pair ($\sigma_{1}\sigma _{2}$) denoting whether the first and second sites are occupied by an oxygen molecule.

(b) The energy of the dimoglobin system can be written as,
E = € 01 +02) + J0102
where $\sigma _{1}$ and $\sigma _{2}$ are the occupation variables of binding site 1 and 2; J is the parameter that describes the cooperativity between the O2 molecules when both states are occupied; $\epsilon$ is the energy associated with an oxygen being bound to one of the two sites.

(c) The Grand Partition function for the dimoglobin system is obtained by adding the weights or Gibbs factor of the four states as shown in the toy model of dimoglobin and is given by,   

Z=1+e-8-) + e-B(6-) + e-8(26+J-24)