where RT = [R] + [RL], the total concentration of free and bound receptors; therefore, [RL]/RT is the fraction of receptors that have a bound ligand. The lower the Kd value, the higher the affinity of a receptor for its ligand. The Kd value is equivalent to the concentration of ligand at which half the receptors contain bound ligand. If [L] = Kd, then from Equation 13-2 we can see that [RL] = 0.5 RT. Equation 13-2 has the same general form as the Michaelis-Menten equation, which describes simple one-substrate enzymatic reactions (Chapter 3). The Kd for a binding reaction is equivalent to the Michaelis constant Km, which reflects the affinity of an enzyme for its substrate.
For a simple binding reaction, Kd = koff/kon, where koff is the rate constant for dissociation of a ligand from its receptor, and kon is the rate constant for formation of a receptor-
ligand complex from free ligand and receptor. The lower koff is relative to kon, the more stable the RL complex, and thus the lower the value of Kd. Like all equilibrium constants, however, the value of Kd does not depend on the absolute values of koff and kon, only on their ratio. For this reason, binding of ligand by two different receptors can have the same Kd values but very different rate constants.
In general, the Kd value of a cell-surface receptor for a circulating hormone is greater than the normal (unstimulated) blood level of that hormone. Under this circumstance, changes in hormone concentration are reflected in proportional changes in the fraction of receptors occupied. Suppose, for instance, that the normal concentration of a hormone in the blood is 10~9 M and that the Kd for its receptor is 10~7 M; by substituting these values into Equation 13-2, we can calculate the fraction of receptors with bound hormone, [RL]/Rt, at equilibrium as 0.0099. Thus about 1 percent of the total receptors will be filled with hormone. If the hormone concentration rises tenfold to 10~8 M, the concentration of receptor-hormone complex will rise proportionately, so that about 10 percent of the total receptors would have bound hormone. If the extent of the induced cellular response parallels the amount of RL, as is often the case, then the cellular responses also will increase tenfold.
In many cases, however, the maximal cellular response to a particular ligand is induced when less than 100 percent of its receptors are bound to the ligand. This phenomenon can be revealed by determining the extent of the response and of receptor-ligand binding at different concentrations of ligand (Figure 13-3). For example, a typical erythroid progenitor cell
Fraction of surface receptors with bound ligand
■ Ligand concentration for 50% physiological response
'Kd for ligand binding
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