  A method is proposed to select the suitable sets of potential parameters for
a one-dimensional mesoscopic Hamiltonian model, first introduced to describe
the DNA melting transition and later extended to investigate thermodynamic and
dynamical properties of nucleic acids. The DNA base pair fluctuations are
considered as time dependent trajectories whose initial condition sets the no
crossing constraint enforced in the path integral for the first-passage
probability. Performing the path integration at room temperature, relations are
established between the cutoff on the amplitude of the base pair fluctuations
and the model parameters. In particular, it is shown that the non-linear
stacking parameter should be $\sim 1$. The formalism here developed may be
applied to compute the lifetime of open base pairs in three-dimensional helical
models for DNA molecules.
