Reversible Binding of a Protein to a Ligand: Oxygen-Binding Proteins:- Cooperative Ligand Binding Can Be Described
Cooperative binding of oxygen by hemoglobin was first analyzed by Archibald Hill in 1910. From this work came a general approach to the study of cooperative ligand binding to multisubunit proteins. For a protein with n binding sites, the equilibrium of Equation 5–1 becomes P+nL⇌PLn (5–12) and the expression for the association constant becomes
The expression for (see Eqn 5–8) is
Rearranging, then taking the log of both sides, yields
where Kd=[L]n 0.5. Equation 5–16 is the Hill equation, and a plot of log [θ/(1-θ)] versus log [L] is called a Hill plot. Based on the equation, the Hill plot should have a slope of n. However, the experimentally determined slope actually reflects not the number of binding sites but the degree of interaction between them. The slope of a Hill plot is therefore denoted by nH, the Hill coefficient, which is a measure of the degree of cooperativity. If nH equals 1, ligand binding is not cooperative, a situation that can arise even in a multisubunit protein if the subunits do not communicate. An nH of greater than 1 indicates positive cooperativity in ligand binding. This is the situation observed in hemoglobin, in which the binding of one molecule of ligand facilitates the binding of others. The theoretical upper limit for nH is reached when nH=n. In this case the binding would be completely cooperative: all binding sites on the protein would bind ligand simultaneously, and no protein molecules partially saturated with ligand would be present under any conditions. This limit is never reached in practice, and the measured value of nH is always less than the actual number of ligand-binding sites in the protein. An nH of less than 1 indicates negative cooperativity, in which the binding of one molecule of ligand impedes the binding of others. Well-documented cases of negative cooperativity are rare. To adapt the Hill equation to the binding of oxygen to hemoglobin we must again substitute pO2for [L] and Pn50 for Kd:
Hill plots for myoglobin and hemoglobin are given in Fig ure 5–14.
FIGURE 5–14 Hill plots for the binding of oxygen to myoglobin and hemoglobin. When nH=1, there is no evident cooperativity. The maximum degree of cooperativity observed for hemoglobin corresponds approximately to nH=3. Note that while this indicates a high level of cooperativity, nH is less than n, the number of O2-binding sites in hemoglobin. This is normal for a protein that exhibits allosteric binding behavior.
