Making Multiple RNA Interaction Practical

Abstract

Multiple RNA interaction can be modeled as a problem in combinatorial optimization, where the “optimal” structure is driven by an energy-minimization-like algorithm. However, the actual structure may not be optimal in this computational sense. Moreover, it is not necessarily unique. Therefore, alternative sub-optimal solutions are needed to cover the biological ground.

We extend a recent combinatorial formulation for the Multiple RNA Interaction problem with approximation algorithms to handle more elaborate interaction patterns, which when combined with Gibbs sampling and MCMC (Markov Chain Monte Carlo), can efficiently generate a reasonable number of optimal and sub-optimal solutions. When viable structures are far from an optimal solution, exploring dependence among different parts of the interaction can increase their score and boost their candidacy for the sampling algorithm. By clustering the solutions, we identify few representatives that are distinct enough to suggest possible alternative structures.

Supported by a Research Starter Award in Informatics from the PhRMA Foundation www.phrmafoundation.org.

Appendix

Given a solution S, define |S| as the number of windows in S, and let

$$w(l_1, l_1',i_1, j_1, u_1, v_1), \ldots , w(l_{|S|}, l_{|S|}', i_{|S|}, j_{|S|}, u_{|S|}, v_{|S|})$$

be the |S| windows in the order defined by the partial order relation follow (from Sect. 2) extended to a total order in a deterministic way.

Each of these windows, say $w(l, l', i, j, u, v)$, defines the two intervals, $[i-u+1,i]$ in level l and $[j-v+1,j]$ in level $l'$. Consider the set of interaction intervals $I(S)=\sum _l I_l(S)$ to be ordered accordingly. Therefore,

$$I(S)=\{I_1,\ldots , I_{2|S|}\} =([i_1-u_1+1, i_1], [j_1-v_1+1, j_1],\ldots $$

$$ \ldots ,[i_{|S|}-u_{|S|}+1, i_{|S|}], [j_{|S|}-v_{|S|}+1,j_{|S|}])$$

is an ordered set of 2|S| intervals. Let $L(S)=\{(l_1,l_1'),\ldots ,(l_{|S|},l_{|S|}')\}$ be an ordered set of |S| pairs, where $(l_i,l_i')$ is the pair defining the $i^{\text {th}}$ window. Therefore, L(S) means that we have the following set of pairwise interactions (not necessarily unique in terms of RNAs): RNA $l_1$ with RNA $l_1'$, RNA $l_2$ with RNA $l_2'$, $\ldots$, RNA $l_{|S|}$ with RNA $l_{|S|}'$. Two solutions that do not agree on this set are considered completely dissimilar; otherwise, their distance is given by the amount of overlap in their interaction intervals (as in the Jaccard metric [21]), hence the following definition of distance:

Given two solutions $S_1$ with $I(S_1)=\{I_1,I_2,\ldots \}$ and $S_2$ with $I(S_2)=\{T_1,T_2,\ldots \}$, the distance between $S_1$ and $S_2$ is

$$d(S_1,S_2)=\left\{ \begin{array}{ccl} 1-\frac{\sum _i |I_i\cap T_i|}{\sum _i |I_i\cup T_i|} &{}\ \ \ &{} L(S_1)=L(S_2)\\ 1 &{}\ \ \ &{} \text {otherwise} \end{array} \right. $$

where $\cap$ and $\cup$ represent the standard intersection and union operations on sets respectively, and intervals are treated as sets of integers.

Recall that a symmetric window $w(l_1,l_2,i,j,u,v)$ satisfies $u=v$ (and typically consists of u base pairs). When applying the distance function, a non-symmetric window is first converted to consecutive symmetric windows by maximizing the number of base pairs (but otherwise is still reported as a non-symmetric window in a given solution).