Dynamic Programming in Computational Biology: RNA Structure & Alignment, Study notes of Algorithms and Programming

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CS 473ug: Algorithms

Mahesh Viswanathan

[email protected]

3232 Siebel Center

University of Illinois, Urbana-Champaign

Spring 2008

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Part I

Dynamic Programming in Computational Biology

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

RNA

Ribonucleic Acid (RNA): Key components in cellular processes

Single-stranded RNA can be viewed as a string of bases: A

(adenine),C (cytosine) ,G (guanine),U (uracil)

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

RNA

Ribonucleic Acid (RNA): Key components in cellular processes

Single-stranded RNA can be viewed as a string of bases: A

(adenine),C (cytosine) ,G (guanine),U (uracil)

Secondary Structure: The single strand loops back to form

bonds between base pairs.

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Secondary Structure

Definition

Secondary structure on an RNA (string) B is a set of pairs

S = {(i, j) | 1 ≤ i, j ≤ n} satisfying

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Secondary Structure

Definition

Secondary structure on an RNA (string) B is a set of pairs

S = {(i, j) | 1 ≤ i, j ≤ n} satisfying

1 [Watson-Crick] No base appears in more than one pair, and

each pair is either {A, U} or {C , G }

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Secondary Structure

Definition

Secondary structure on an RNA (string) B is a set of pairs

S = {(i, j) | 1 ≤ i, j ≤ n} satisfying

1 [Watson-Crick] No base appears in more than one pair, and

each pair is either {A, U} or {C , G }

2 [No sharp turns] Ends of each pair are separated by distance

at least 4, i.e., i < j − 4

3 [Non-crossing] If (i, j) and (k, `) are pairs then we cannot

have i < k < j < `

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Examples of Secondary Structure

A

U

G

G

G

G

C

A

U

A U G G G G C A U

Figure: Has sharp turn

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Examples of Secondary Structure

A

U

G

G

G

G

C

A

U

A U G G G G C A U

Figure: Has sharp turn

A

G

U

U

G G

C

C

A

U

A G U U G G C C A U

Figure: Has crossing

A

U

G

U

G G

C

C

A

U

A U G U G G C C A U

Figure: Correct pairing

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Free Energy

Example

Consider the RNA molecule AGGAUCGCCU

A

G

G

A

C U

G

C

C

U

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Free Energy

Example

Consider the RNA molecule AGGAUCGCCU

A

G

G

A

C U

G

C

C

U A

G

G

A

C U

G

C

C

U

Which structure is exhibited?

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Free Energy

Example

Consider the RNA molecule AGGAUCGCCU

A

G

G

A

C U

G

C

C

U A

G

G

A

C U

G

C

C

U

Which structure is exhibited?

Observation

RNA assumes the secondary structure of minimum free energy.

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

The Problem

Free Energy Model

The larger the number of base pairs, the lower the energy.

Computational Problem

Input An RNA molecule b 1 b 2... bn

Goal Find a set a base pairs of maximum size that follows

the Watson-Crick rules, does not have sharp turns,

and in the base pairs don’t cross each other.

RNA Secondary Structure Sequence Alignment Space Efficient Sequence Alignment

Secondary Structure Computational Problem Dynamic Programming Solution

Defining Subproblems

For b 1... bn, let Opt(j) be the maximum number of base pairs in

b 1... bj

Case bn is not paired in secondary structure.

Case bn is paired with bi in optimal pairing.