Complexity and Tractability. Statistical Mechanics of Helix-Coil Transitions in Circular DNA as a Model-Problem

A general class of one-dimensional, sequence-specific, Ising models is defined, with length-dependent long-range effects extending throughout the lengths, n, of the systems. The models are relevant to helix-coil transitions in DNA molecules. With k, mutually coupled, long-range effects the models provide a simple prototype for intractable problems, with algorithmic time complexities growing in O(nk + 1). It is shown that there is a single rigorous solution which reduces the complexities of the models from O(nk + 1) to O(n), with the long-range effects expressed as multiexponential functions. Using this method, partition functions may be evaluated much faster than with an exact treatment. The calculations are performed with prescribed numerical accuracy, following the accuracy of the multiexponential representations of the long-range effects. For relevant classical models (with k = 2 or 3, and n = 5000), the calculation times are reduced by factors ranging between thousands and millions, with accuracy better than 1%.