  We construct and study an extended random matrix model of RNA (polymer)
folding. A perturbation which acts on all the nucleotides in the chain is added
to the action of the RNA partition function. The effect of this perturbation on
the partition function and the Genus Distributions is studied. This
perturbation distinguishes between the paired and unpaired bases. For example,
for $\alpha = 1$ (where $\alpha$ is the ratio of the strengths of the original
and perturbed term in the action) the partition function and genus distribution
for odd lengths vanish completely. This partition function and the genus
distribution is non-zero for even lengths where structures with fully paired
bases only remain. This implies that (i). the genus distributions are different
and (ii). there is a ``structural transition'' (from an ``unpaired-paired base
phase'' to a ``completely paired base phase'') as $\alpha$ approaches 1 in the
extended matrix models. We compare the results of the extended RNA model with
the results of G. Vernizzi, H. Orland and A. Zee in PRL 94, 168103(2005).
