  We show how single-molecule unzipping experiments can provide strong evidence
that the zero-force melting transition of long molecules of natural dsDNA
should be classified as a phase transition of the higher-order type
(continuous). We study a model for a long molecule of dsDNA, and compute the
equilibrium phase diagram for the experiment in which the molecule is unzipped
under force. We consider a perfect-matching dsDNA model, in which the loops are
volume-excluding chains with arbitrary loop exponent c. We include stacking
interactions, hydrogen bonds, and main-chain entropy, including sequence
heterogeneity at the level of random sequences. We use the replica method to
calculate the equilibrium properties of the system. As a function of
temperature, we obtain the minimal force at which the molecule separates
completely. This critical force curve is a line in the temperature-force phase
diagram that marks the regions where the molecule exists primarily as a helix,
versus the region where the molecule exists as two separate strands. Near
melting, the critical force curve of our random-sequence model is very
different from that of the homogeneous version of our model. For both sequence
models, the critical force falls to zero at the melting temperature with a
power law having exponent alpha. For the homogeneous model, alpha is 1/2 almost
exactly, while for the random model, alpha is about 0.9. The shape of the
critical force determines how the helix fraction falls to zero at melting, and
thus classifies the melting transition as a type of phase transition.
