Saturation functions as a nested set.

In a recent paper, Bardsley & McGinlay (1987) use saturation functions for binding of a ligand to a protein with an unknown number of binding sites to explore the power and reliability of the F test when used outside its range of strict validity. In the course of their discussion they make the interesting and novel observation that binding equations do not form a nested set: increasing the number of terms does not give a model that includes the simpler ones as special cases, and does not necessarily result in a minimum sum of squares smaller than or equal to that obtained with an equation with fewer terms. In their Appendix 5 they prove that this is a real result rather than, for example, a computing artifact. There is an important sense, however, in which saturation functions do constitute a nested set. For convenience, one often treats the fractional saturation y, defined as the number of occupied binding sites divided by the total number of binding sites whether occupied or not, as the dependent variable, and this is what Bardsley & McGinlay (1987) do in their paper. However, y is never measured directly; it must be calculated by dividing the observed degree of binding by the degree of binding at saturation. This limit is commonly obtained in either of two ways which differ in their consequences for model testing. One way is to estimate the limit by extrapolation of the observed behaviour to infinite ligand concentration, and in this case the value ofy is model-independent and it is quite proper to treat it as dependent variable in the statistical calculations. (Nonetheless, one might prefer to treat the observed degree of binding as the dependent variable and to include the limit as a parameter to be estimated.) Alternatively, if the purity and molecular mass of the protein are known it is preferable to take as dependent variable the average number of ligand molecules bound per molecule of protein, N~: this corresponds to ny, where n is the number of binding sites per molecule. In this case y is model-dependent, and when testing the significance of including an extra term in the binding equation, therefore, i.e. increasing n from i to i+ 1, one must not keep all the y values unchanged but must divide them by (1 + I/i). One can avoid this complication by taking N~ explicitly as dependent variable …