  We study a matrix model of RNA in which an external perturbation acts on n
nucleotides of the polymer chain. The effect of the perturbation appears in the
exponential generating function of the partition function as a factor
$(1-\frac{n\alpha}{L})$ [where $\alpha$ is the ratio of strengths of the
original to the perturbed term and L is length of the chain]. The asymptotic
behaviour of the genus distribution functions for the extended matrix model are
analyzed numerically when (i) $n=L$ and (ii) $n=1$. In these matrix models of
RNA, as $n\alpha/L$ is increased from 0 to 1, it is found that the universality
of the number of diagrams $a_{L, g}$ at a fixed length L and genus g changes
from $3^{L}$ to $(3-\frac{n\alpha}{L})^{L}$ ($2^{L}$ when $n\alpha/L=1$) and
the asymptotic expression of the total number of diagrams $\cal N$ at a fixed
length L but independent of genus g, changes in the factor $\exp^{\sqrt{L}}$ to
$\exp^{(1-\frac{n\alpha}{L})\sqrt{L}}$ ($exp^{0}=1$ when $n\alpha/L=1$)
