Theory of the Helix—Coil Transition for Synthetic Polynucleotides Forming Branched Helical Structures

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Abstract
A theory is developed for synthetic polynucleotides such as dAT, which are capable of forming single‐stranded, branched helical structures. The grand‐partition function for a dilute solution is expressed in terms of the so‐called quasigrand partition function for a strand. Configurations are considered which consist of rings in the random‐coil state, connected by helices. Weights are assigned to each configuration according to a similar set of rules as were used by Crothers and Zimm [D. M. Crothers and B. H. Zimm, J. Mol. Biol. 9, 1 (1964)] for unbranched two‐stranded complexes. The enumeration of configurations is carried out by means of a graph—theoretical technique. Expressions are derived for the helicity (the fraction bonded pairs), the mean number of base pairs per helix and per ring, the mean numbers of helices and U loops per strand. Our results for self‐complementary strands (dAT) are compared with the results of Crothers and Zimm for pairs of complementary homopolymer strands (dG—dC). The slope of our melting curve is found to be proportional to the ½ power of the stacking parameter τ instead of the ⅔‐power dependence obtained for unbranched systems.