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.

The matrix model of (simplified) RNA folding with an external linear interaction in the action of the partition function is reviewed. The important results for structure combinatorics of the model are discussed and analysed in terms of the already existing models.

The Penner interaction known in studies of moduli space of punctured Riemann surfaces is introduced and studied in the context of random matrix model of homo RNA. An analytic derivation of the generating function is given and the corresponding partition function is derived numerically. An additional dependence of the structure combinatorics factor on 𝑁 (related to the size of the matrix and the interaction strength) is obtained. This factor has a strong effect on the structure combinatorics in the low 𝑁 regime. Databases are scanned for real ribonucleic acid (RNA) structures and pairing information for these RNA structures is computationally extracted. Then the genus is calculated for every structure and plotted as a function of length. The genus distribution function is compared with the prediction from the nonlinear (NL) model. The specific heat and distribution of structure with temperature calculated from the NL model shows that the NL inter-action is biased towards planar structures. The second derivative of specific heat changes phase from a double peaked function for small 𝑁 to a single peak for large 𝑁. Detailed analysis reveals the presence of the double peak only for genus 0 structures, the higher genii behave normally with 𝑁. Comparable behaviour is found in studies involving interactions of RNA with osmolytes and monovalent cations in unfolding experiments.