Derivation - Cooperativity results from a conformational "switch" from a weak-binding state to a strong-binding state. This switching is not easily visualized when binding curves are represented as in Figure 7.8c or d, nor do such curves give an easy way to measure the degree of cooperativity. Rearranging Equation 7.6, yields

/(1 - ) = PO₂/P50,

where is the fraction of oxygen-binding sites that are occupied, PO₂ is the partial pressure of oxygen, and P50 is the oxygen partial pressure for half-saturation.

Taking logarithms of both sides,

Graphing log [ /(1 - )] versus log PO₂ produces what is called a Hill plot (Figure 7.9). The Hill plot for noncooperative binding will, according to equation (7.9), be a straight line with slope = 1. In such a plot, the abscissa value (the value of log PO₂) corresponding to log [/(1 - )] = 0 will equal log P50. When hemoglobin first begins binding (at low PO₂), its Hill plot has a slope >1, corresponding to the weak-binding state (large P50). As binding progresses, the curve switches over to approach another, parallel line that describes the strong-binding state (small P50).

Advantages of Hill Plots - Hill plots readily identify the transition between binding states in cooperativity, and the binding behavior is unmistakably different for cooperative and noncooperative systems. Furthermore, the Hill plot gives a direct numerical measure of the degree of cooperativity from its maximum slope, nH, which is called the Hill coefficient.

Interpretating Plots - Three cases may be considered for a molecule with n binding sites:

1. nH = 1: The molecule binds noncooperatively. This situation can happen even with a multisite protein if the sites do not communicate with one another.

2. 1 < nH < n: This situation is the usual one for a cooperatively binding protein, as depicted in Figure 7.9. The Hill coefficient must be greater than unity in order for the Hill plot to switch over from the weak-binding line to the strong-binding line.

3. nH = n: In this hypothetical situation the molecule is wholly cooperative. In such a situation, one hemoglobin molecule would fill up its four oxygen-bindingsites before any others had taken oxygen, so that only wholly unliganded and wholly liganded molecules would be present at any point in the binding process. If this were the case, the binding curve would have the form

and the Hill equation would become

which is a straight line with slope = n. This case is never seen in reality. For example, the Hill coefficient of hemoglobin (n = 4) never exceeds about 3.5.

Allosteric Effects - The cooperative binding of oxygen by hemoglobin is one example of what is referred to as allosteric effects. In allosteric binding, the uptake of one ligand by a protein influences the affinities of remaining unfilled binding sites. The ligands may be of the same kind (as in hemoglobin), or they may be different as in the the way binding of 2,3-bisphosphoglycerate to hemoglobin affects the protein's affinity for oxygen (see here). Allostery is also an important mechanism for regulating the activity of enzymes. For example, both the enzymatic activity and the substrate preferences of the nucleotide metabolism enzyme, ribonucleotide reductase, are controlled by small effector molecules, such as ATP (see here). In that case, allostery allows one kind of small molecule to regulate the action of a protein on another kind of molecule. The ability of multisubunit proteins to be regulated allosterically may be one of the reasons these proteins are so common.