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Formulas for mean restriction-fragment lengths and related quantities.

Bishop et al. (1983) presented a model for the study of the fragment-length distribution produced when DNA is digested using a restriction enzyme. They discuss the value of such a model and derive, as their main result, the mean fragment length produced by complete digestion with a restriction enzyme having a unique recognition site. From this they obtain (1) an approximation to the mean fragment length when more than one enzyme is used simultaneously (or, equivalently, when one enzyme recognizes more than one sequence) and (2) the relative frequency of the different types of fragments produced, which involves the iteration of a certain matrix. The method used by Bishop et al. (1983) yields appropriate numerical answers to their main problem. The purpose of the present paper is to demonstrate how one can obtain both fairly simple formulas for the mean fragment lengths and other quantities of interest. Evaluation of these, if required, would be trivial, and the iteration of what can be quite large matrices is avoided. The extension in Bishop et al. (1983) from one enzyme to many involves an approximation-essentially justified by the fairly long (in number of bases) fragments produced-that will be reasonable for most problems connected with DNA digestion. The results obtained in the present paper are exact, and these and some others are derived by Biggins and Cannings (1987) using the theory of semi-Markov processes. We believe the approach used here is more accessible to the nonmathematician and of some independent interest. The main value of the method presented here, from the geneticist's point of view, lies in its relative simplicity.