In the matrix model of RNA [G Vernizzi, H Orland and A Zee, Phys. Rev. Lett. 94, 168103 (2005)] we introduce external interactions on n bases in the action of the partition function where $n \leq L$ and 𝐿 is the length of the polymer chain. The RNA structures found in the model can be separated into two regimes: (i) $0 \leq \alpha \leq 1$, $n < L$ and $0 \leq \alpha < 1$, $n = L$ where unpaired and paired base structures exist and (ii) $\alpha = 1$, $n = L$ with only completely paired base structures. Figures for the genus distribution functions show differences at small lengths. We consider the situation when the strength of external perturbation is different on different bases in the polymer chain.