Asymptotic Number of Hairpins of Saturated RNA Secondary Structures

Abstract

In the absence of chaperone molecules, RNA folding is believed to depend on the distribution of kinetic traps in the energy landscape of all secondary structures. Kinetic traps in the Nussinov energy model are precisely those secondary structures that are saturated, meaning that no base pair can be added without introducing either a pseudoknot or base triple. In this paper, we compute the asymptotic expected number of hairpins in saturated structures. For instance, if every hairpin is required to contain at least θ=3 unpaired bases and the probability that any two positions can base-pair is p=3/8, then the asymptotic number of saturated structures is 1.34685⋅n ^−3/2⋅1.62178ⁿ, and the asymptotic expected number of hairpins follows a normal distribution with mean $0.06695640 \cdot n + 0.01909350 \cdot\sqrt{n} \cdot\mathcal{N}$. Similar results are given for values θ=1,3, and p=1,1/2,3/8; for instance, when θ=1 and p=1, the asymptotic expected number of hairpins in saturated secondary structures is 0.123194⋅n, a value greater than the asymptotic expected number 0.105573⋅n of hairpins over all secondary structures. Since RNA binding targets are often found in hairpin regions, it follows that saturated structures present potentially more binding targets than nonsaturated structures, on average. Next, we describe a novel algorithm to compute the hairpin profile of a given RNA sequence: given RNA sequence a ₁,…,a _n , for each integer k, we compute that secondary structure S _k having minimum energy in the Nussinov energy model, taken over all secondary structures having k hairpins. We expect that an extension of our algorithm to the Turner energy model may provide more accurate structure prediction for particular RNAs, such as tRNAs and purine riboswitches, known to have a particular number of hairpins. Mathematica^™ computations, C and Python source code, and additional supplementary information are available at the website http://bioinformatics.bc.edu/clotelab/RNAhairpinProfile/.

Appendix: Computing the Number of Hairpins in Saturated Structures

To produce Fig. 1, we computed by dynamic programming the expected number of hairpins in saturated structures for a homopolymer of size n. In the interests of brevity, we must refer the interested reader to Clote (2006) for background material on recurrence relations for the number of saturated structures. The recurrence relations require the auxiliary notion of saturated structure with no visible positions, defined as follows. A secondary structure S on sequence a ₁,…,a _n has no visible positions, if for all 1≤i≤n in which a _i is unpaired, there is no base pair (x,y) for which x<i<y.

Let D(n,k) denote the number of saturated secondary structures having exactly k hairpins. Let E(n,k) denote the number of saturated secondary structures having exactly k hairpins, which have no visible positions. Define D(0,0)=D(1,0)=D(2,0)=D(3,0)=1 and E(0,0)=E(3,1)=1; for all other values of 0≤n≤3 and 0≤k≤3, let D(n,k)=E(n,k)=0.

The inductive case is given by:

$$\begin{aligned} D(n,k) =&E(n-1,k)+E(n-2,k) + \sum_{r=1}^{n-2} D(r-1,k-1)D(n-r-1,0) \\ &{}+ \sum_{r=1}^{n-2} \sum _{s=0}^{k-1} D(r-1,s)D(n-r-1,k-s) \\ E(n,k) =& \sum_{r=1}^{n-2} E(r-1,k-1)D(n-r-1,0) \\ &{} +\sum_{r=1}^{n-2} \sum _{s=0}^{k-1} E(r-1,s)D(n-r-1,k-s). \end{aligned}$$

Since the justification for these recursion is similar to that of Clote (2006), we do not provide further details. These recursions are implemented using dynamic programming to compute the number of saturated structures on a homopolymer of size n having exactly k hairpins. It follows that the expected number of hairpins for a homopolymer of size n is

$$\sum_{k=0}^n k \cdot\frac{D(n,k)}{S(n)} $$

where $S(n)=\sum_{k=0}^{n} D(n,k)$ is the total number of saturated structures for a homopolymer of size n. The Python code is available on the web supplement.

Definition of Resultant

In the proof of Theorem 3, we compute the resultant of two multivariable polynomials. For the benefit of the reader, we define this concept here. For any commutative ring A, indeterminate X and two multivariate polynomials

$$\begin{aligned} p_1 =& v_n X^n + \cdots+ v_1 X + v_0 \\ p_2 =& u_m X^m + \cdots+ u_1 X + u_0 \end{aligned}$$

respectively having roots α ₁,…,α _n and β ₁,…,β _m in the algebraic closure of A, the resultant of p ₁,p ₂ with respect to X is defined to be

$$v_n^n u_m^m \prod _{i=1}^n \prod_{j=1}^m (\alpha_i-\beta_j). $$

In applications, for instance g ₁,g ₂ could be functions in variables S,R,u,z, but construed to be polynomials over indeterminate R with coefficients from the ring $\mathbb{Z}(z,u,S)$. In such a case, the resultant Res(g ₁,g ₂) of g ₁,g ₂ is a polynomial in $\mathbb{Z}[z,u,S]$, whose roots are the z-, u- and S-coordinates of the intersection of curves corresponding to g ₁,g ₂. Moreover, it is known that there exist polynomials $q_{1},q_{2} \in\mathbb{Z}[z,u,S][R]$ such that

$$ g_1 \cdot q_1 + g_2 \cdot q_2 = \mathit{Res}(g_1,g_2). $$

(7)

For more background on resultants, see Lang (2002).