Geometric Combinatorics and RNA Folding Algorithms

Geometric combinatorics and RNA folding algorithms

Qijun He (Clemson University)

Joint work with: Christine Heitsch (Georgia Tech) Svetlana Poznanovikj∗ (Clemson) Andrew Gainer-Dewar∗ (UConn Health Center) Elizabeth Drellich (North Texas) Heather Harrington (Oxford)

Mathematics Research Communities (MRC) 2014 Snowbird, Utah

ACSB Conference University of Connecticut Health Center May 23, 2015

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms Introduction

I am a 4th-year PhD student at Clemson University, interested in combinatorics, computational algebra, and mathematical biology.

Current research projects Applying geometric combinatorics (polytopes) and parametric inference to RNA folding. [What I’ll talk about today] With MRC 2014 research group led by Christine Heitsch & Svetlana Poznanovikj. k-canalizing Boolean functions. With Matthew Macauley & Elena Dimitrova. [See my poster!] Data identiﬁcation using Gr¨obnernormal forms. With Elena Dimitrova & Brandy Stigler.

Advertisments I am co-organizing (with Raina Robeva & Andy Jenkins) a mini-symposium on discrete and algebraic biology at the 2015 Biomathematics and Ecology Education and Research (BEER) Symposium in October 2015. I am the organizing chair of the 12th Graduate Students Combinatorics Conference which will be at Clemson in March 2016! I will be on the job market next year. Know of anyone looking for a postdoc?

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms What is RNA?

RNA( Ribonucleic acid) are biological molecules built from strings of nucleotides.

adenine (A), guanine (G), cytosine (C), thymine (T), uracil (U)

RNA strands consist of A, C, G, and U. Combinatorially, an RNA strand is a length-n sequence, over the alphabet {A, C, G, U}.

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms RNA sequences via base pairings

Primary sequence −→ Secondary structure −→ 3D molecule

RNA secondary structures balance energetically favorable helices (consecutive base pairs) against destabilizing loops (single-stranded bases).

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms Secondary structure & pseudoknots

Here are two folds of the same RNA strand, and the corresponding arc diagrams. The ﬁrst is a secondary structure and the second is a pseudoknot.

50-end G G G A C C U C C C C C C U A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 A G G G G G 30-end G G G A C C U U C C C C C C A A G G G G G G G G G

50-end G G G A C C U U C C C C C C A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 G A G G G G G G 30-end G G G A C C U U C C C C C C A A G G G G G G G

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms Secondary structure prediction

The problem For a given sequence, there are many possible secondary structures into which it can fold. What is the most ‘likely’ one?

Minimal free energy (mfe) model The optimal secondary structure minimizes the free energy, ∆G.

Example energy model

Given an RNA sequence S = b1b2 ··· bn, let  −3 {bi , bj } = {C, G} and i ≤ j − 4  −2 {b , b } = {A, U} and i ≤ j − 4 δg(i, j) = i j −1 {bi , bj } = {G, U} and i ≤ j − 4  0 otherwise.

be the free energy of the potential bond between bi and bj . Find the structure that minimizes ∆G, the sum of the energies of the base pairs.

This can be done using dynamic programming (DP) to recurse on the substructures.

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms MFE folding: toy example

There are 4 ways to recurse on the substructure Si,j = bi bi+1 ··· bj−1bj .

These correspond to the following 4 cases about how bases bi and bj can bond:

◦◦◦ ... ◦◦◦ ◦◦...◦◦◦ ... ◦◦ ◦◦ ... ◦◦... ◦◦◦ ◦◦ ... ◦◦...◦... ◦◦... ◦◦

i j i i +1 k j−1 j i i +1 k j−1 j i i +1 ki k kj j−1 j

Thus the optimal energy score ∆G(i, j) of subsequence Si,j is given by:  ∆G(i + 1, j − 1) + δg(i, j)  ∆G(i + 1, j) ∆G(i, j) = min ∆G(i, j − 1)   min ∆G(i, k) + ∆G(k + 1, j). i

Our ﬁnal goal is to compute S1,n.

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms A toy example: S = GGGACCUUCC

j G G G A C C U U C C G G G A C C U U C C G 0 0 0 0 −3 G 0 0 0 0 −3−3−4−4−6−8 G 0 0 0 0 −3 G 0 0 0 0 −3−3−3−5−8 G 0 0 0 0 −1 G 0 0 0 0 −1−2−5−5 A 0 0 0 0 −2 A 0 0 0 0 −2−2−2 C 0 0 0 0 0 C 0 0 0 0 0 0 i C 0 0 0 0 0 C 0 0 0 0 0 U 0 0 0 0 U 0 0 0 0 U 0 0 0 U 0 0 0 C 0 0 C 0 0 C 0 C 0

Figure: Recording the optimal scores in a table during a DP routine.

G G G A C C U U C C

∆G(S) = −8

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms MFE folding: toy example

We can use the language from OR to describe RNA folding folding problems.

In our toy example of S = GGGACCUUCC, the problem can be rephrased as:

min ∆G = −3k1 − 2k2 − k3,

such that, there exist a ‘valid’ structure T on S with

k1 CG pairs,

k2 AU pairs,

k3 GU pairs.

We know little about the feasible region of this optimization problem, but we know it is ﬁnite.

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms NNTM and GTfold

Nearest neighbor thermodynamic model: linear objective function with over 7,000 parameters (Turner99 database)!

GTfold: DP algorithm based on NNTM (using Turner99 database) to minimize the free energy and provide optimal RNA folding structure.

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms NNTM and GTfold

DP solves discrete optimization eﬃciently, but quality of free energy approximation by the NNTM objective function varies widely.

What might go wrong?

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms Multibranch loop parameters

For computational eﬃciency, only3 parameters are used to govern multibranch loops. Even worse: they are almost purely ‘made up’.

Multibranch loop parameters a: energy penalty for a multibranch loop. b: energy for each unpaired nucleotide in a multibranch loop. c: energy for each branching helix in a multibranch loop.

Question How do multibranch loop parameters (a, b, c) aﬀect the optimal structure?

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms Multibranch loop parameters

We want to run parametric analysis on (a, b, c), but we don’t know how ‘wrong’ they are.

Answer? We need to construct the convex hull (a polytope) of the feasible region.

The optimal value should exist on an extreme point of the feasible region (a vertex of this polytope).

The real problem The dimension of this polytope is over 7,000 and we know little about the feasible region.

Qijun He (Clemson) Geometric combinatorics and RNA folding algorithms Dimension reduction

For a given structure T , we can write its free energy as:

∆G(T ) = axT + byT + czT + wT , where

xT : number of multibranch loops in T ,

yT : number of unpaired nucleotides in multibranch loops in T ,

zT : number of branching helices in multibranch loops in T ,

wT : energy of the remaining structures using Turner99 parameters.

The proﬁle space is (xT , yT , zT , wT ), and we introduce a dummy variable d:

∆G(T ) = axT + byT + czT + dwT .

Question How do multibranch loop parameters (a, b, c, d) aﬀect the optimal structure?

The real problem How to compute the convex hull of an implicit ﬁnite feasible region?

Answer! Let GTfold do it for us!