Unified approach to partition functions of RNA secondary structures

Abstract

RNA secondary structure formation is a field of considerable biological interest as well as a model system for understanding generic properties of heteropolymer folding. This system is particularly attractive because the partition function and thus all thermodynamic properties of RNA secondary structure ensembles can be calculated numerically in polynomial time for arbitrary sequences and homopolymer models admit analytical solutions. Such solutions for many different aspects of the combinatorics of RNA secondary structure formation share the property that the final solution depends on differences of statistical weights rather than on the weights alone. Here, we present a unified approach to a large class of problems in the field of RNA secondary structure formation. We prove a generic theorem for the calculation of RNA folding partition functions. Then, we show that this approach can be applied to the study of the molten-native transition, denaturation of RNA molecules, as well as to studies of the glass phase of random RNA sequences.

Appendix A: Molten partition function for even length

In this appendix we calculate the $z$-transform of the molten phase transition function for even sequence length only. For clarity we suppress the argument $q$ in all partition functions and understand that all partition functions depend on the Boltzmann factor $q$.

We will actually calculate the $z$-transforms for even and odd sequence lengths simultaneously and start by defining

$$\begin{aligned} E(N)\equiv G(2N-1)\quad {\mathrm{and}}\quad O(N)\equiv G(2N) \end{aligned}$$

for $N\ge 1$ (note, that odd arguments of $G$ correspond to chains of an even length and vice versa). Here, $G(N)$ is the molten phase partition function introduced in Sect. 2.4. These quantities have $z$-transforms

$$\begin{aligned} \widehat{E}(z)\equiv \sum _{N=1}^\infty E(N)z^{-N}\quad {\mathrm{and}}\quad \widehat{O}(z)\equiv \sum _{N=1}^\infty O(N)z^{-N}. \end{aligned}$$

By their construction we have the relation

$$\begin{aligned} z\widehat{E}(z^2)+\widehat{O}(z^2)=\!\!\!\sum _{N=1}^\infty G(2N-1)z^{-(2N-1)} +\!\sum _{N=1}^\infty G(2N)z^{-2N}\!=\widehat{G}(z).\qquad \quad \end{aligned}$$

(23)

Let us additionally insert $N=2n-1$ for $n\ge 2$ into Eq. (5). This gives us

$$\begin{aligned} O(n)&= E(n)+q\sum _{k=1}^{2n-2}G(k)G(2n-1-k)\\&= E(n)+q\sum _{k=1}^{n-1}G(2k-1)G(2n-2k)+q \sum _{k=1}^{n-1}G(2k)G(2n-2k-1)\\&= E(n)+2q\sum _{k=1}^{n-1}G(2k-1)G(2n-2k) =E(n)+2q\sum _{k=1}^{n-1}E(k)O(n-k). \end{aligned}$$

Multiplying this equation by $z^{-n}$, summing over all $n\ge 2$ and adding $z^{-1}$ on both sides yields

$$\begin{aligned} \widehat{O}(z)=\widehat{E}(z)+2q\widehat{E}(z)\widehat{O}(z) \end{aligned}$$

which in turn can be rewritten as

$$\begin{aligned} \widehat{O}(z)=\frac{\widehat{E}(z)}{1-2q\widehat{E}(z)}. \end{aligned}$$

(24)

If we insert this into Eq. (23) we get the equation

$$\begin{aligned} \widehat{G}(\sqrt{z})=\sqrt{z}\widehat{E}(z) +\frac{\widehat{E}(z)}{1-2q\widehat{E}(z)} \end{aligned}$$

for $\widehat{E}(z)$. This is a quadratic equation for $\widehat{E}(z)$. Inserting the explicit expression (8) for $\widehat{G}(z)$ this quadratic equation can be solved explicitly with the result

$$\begin{aligned} \widehat{E}(z)=\frac{1}{2q}-\frac{1}{4q\sqrt{z}}\left[ \sqrt{(\sqrt{z}-1)^2-4q}+\sqrt{(\sqrt{z}+1)^2-4q}\right] . \end{aligned}$$

(25)

The other solution with a minus sign between the two square roots has to be excluded, since $\widehat{O}(z)>0$ and $\widehat{E}(z)>0$ for real and positive $z$ and thus, according to Eq. (24), $\widehat{E}(z)<1/(2q)$.