Occurrence of a Phase Transition in Nucleic Acid Models

Abstract

The double‐stranded helix found in nucleic acids suggests a model where order is represented by specific, one‐to‐one bonding between two infinite chains and where disorder arises from the formation of large loops. The statistical weight per unit in an unordered sequence (i.e., a loop) is given by (1/i) lnui =a—(1/i) (b+clni). This quantity becomes large, as i is increased, slower than 1/i because of the c lni term. From the previous paper, this suggests that a first‐order phase transition may take place. It is found that this happens if c>2, in which case both U(x) (the sequence generating function for loops) and ∂U(x)/∂x converge at the point where x 1 (the unit partition function) equals u 2 (the statistical weight per unit of an infinite loop, i.e., a free chain). If 2>c>1, then U(x) converges at x 1=u 2 but ∂U(x)/∂x does not. For a loop in three dimensions, c would be about 3 2 (½ for each dimension); hence, there is no discontinuity in θ, the fraction of ordered states, for a real system. However, θ does behave nonanalytically at x 1=u 2, giving a continuous or higher‐order transition. Specifically, θ does go exactly to zero in the transition region. This is a qualitative difference from infinite polypeptides where the sigmoidal transition curve goes asymptotically to 1 and 0 at the extremes of temperature. Explicit calculations are given for various values of c to illustrate the behavior of the model.