RNA Secondary Structure

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Predicting RNA Secondary Structures A Lattice
Walk Approach to Modeling Sequences Within the
HIV-1 RNA Structure

• Facing the Challenge of Infectious Diseases in
  Africa The Role of Mathematical Modeling
• University of Witswatersrand
• Johannesburg, South Africa
• September 25-27, 2006
• Asamoah Nkwanta, Ph.D.
• Morgan State University
• Nkwanta_at_jewel.morgan.edu

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TOPICS

• RNA Prediction Molecular Biology
• RNA Combinatorics
• Certain Class of Random Walks
• Matrix Theory
• Connection Between Walks RNA
• Modeling HIV-1 RNA Sequences

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RNA Secondary Structure Prediction
The Human Genome Project and related efforts
have generated enormous amounts of raw biological
sequence data. However, understanding how
biological sequences encode structural
information remains a fundamental scientific
challenge. For instance, understanding the base
pairing, or secondary structure, of
single-stranded RNA sequences is crucial to
advancing knowledge of their novel biochemical
functions. C. E. Heithsch, Combinatorics
on Plane Trees, Motivated by RNA Secondary
Structure Configuration (preprint, 2005)
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What is RNA Secondary Sequence Prediction ?
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RNA Secondary Structure Prediction

• Given a primary sequence, we want to find the
  biological function of the related secondary
  structure. To achieve this goal we predict
  (model) its secondary structure.
• Most methods predict secondary structure rather
  than tertiary structure. The three dimensional
  shape is important for biological function, and
  it is harder to predict.

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Molecular Biology (Cont.)
3-D structure of Haloarcula marismortui 5S
ribosomal RNA in large ribosomal subunit
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Molecular Biology

• Central Dogma
• DNA ? RNA ? Protein
• Transcription / Translation

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Molecular Biology (Cont.)
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Molecular Biology (Cont.)
However, the "Central Dogma" has had to be
revised a bit.  It turns out that you CAN go back
from RNA to DNA, and that RNA can also make
copies of itself.  It is still NOT possible to go
from Proteins back to RNA or DNA, and no known
mechanism has yet been demonstrated for proteins
making copies of themselves.
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Molecular Biology (cont.)

• HIV is one of a group of atypical viruses called
  retroviruses that maintain their genetic
  information in the form of RNA. Retroviruses are
  capable of producing DNA from RNA.

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Molecular Biology (Cont.)
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Molecular Biology (cont.)

• Ribonucleic acid (RNA) molecule Three main
  categories
• mRNA (messenger) carries genetic information
  from genes to other cells
• tRNA (transfer) carries amino acids to a
  ribosome (cells for making proteins)
• rRNA (ribosomal) part of the structure of a
  ribosome

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Molecular Biology (cont.)

• Other types (RNA) molecules
• snRNA (small nuclear RNA) carries genetic
  information from genes to other cells
• miRNA (micro RNA) carries amino acids to a
  ribosome (cells for making proteins)
• iRNA (immune RNA) part of the structure of a
  ribosome (Important for HIV studies)

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RNA Secondary Structure

• RNA secondary structures are important in many
  biological processes and efficient structure
  prediction can give vital directions for
  experimental investigation.
• B. Knudsen and J. Hein, Pfold RNA secondary
  structure prediction using stochastic
  context-free grammars (Nucleic Acids Research,
  2003)
• There are published examples involving tRNA,
  rRNA, and other types of RNA

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RNA Secondary Structure (cont.)

• A ribonucleic acid (RNA) molecule consists of a
  sequence of ribonucleotides (typically single
  stranded)
• Each ribonucleotide contains one of four bases
  adenine (A), cytosine (C), guanine (G), and
  uracil (U)

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Secondary Structure (cont.)

• Note U is replaced by thymine (T) in DNA
• As the molecule forms, chemical bonds join A-U
  and C-G pairs, (Unstable G-U). These are called
  the Watson-Crick pairs.

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Secondary Structure (cont.)

• Primary Structure The linear sequence of bases
  in an RNA molecule
• Secondary Structure The folding or coiling of
  the sequence due to bonded nucleotide pairs A-U,
  G-C
• Tertiary Structure The three dimensional
  configuration of an RNA molecule

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Primary RNA Sequence

• CAGCAUCACAUCCGCGGGGUAAACGCU
• Nucleotide Length, 27 bases

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Geometric Representation

• Secondary structure is a graph defined on a set
  of n labeled points
• (M.S. Waterman, 1978)
• Biological
• Combinatorial/Graph Theoretic
• Random Walk

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RNA COMBINATORICS

• RNA Numbers 1,1,1,2,4,8,17,37,82,185,423,978,
• These numbers count various combinatorial objects
  including RNA secondary structures of length n.

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RNA COMBINATORICS (cont.)

• The number of RNA secondary structures for the
  sequence 1,n is counted by the coefficients of
  s(z)
• Coefficients of the formal power series
• (1,1,1,2,4,8,17,37,82,185,423,978,)

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RNA COMBINATORICS (cont.)

• The number of lattice paths with unit steps R
  (right), U (up) D (down) that go from (0,0),
  remain in the first quadrant of the coordinate
  plane, and return to the x-axis under the
  restriction that there are never consecutive UD
  steps is the nth RNA number
• (1,1,1,2,4,8,17,37,82,185,423,978,)

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RNA COMBINATORICS (cont.)

• The number of RNA sequences of length n that can
  be formed over the words A,U,G,C such that the
  letters A U are not adjacent is equal to
• What a remarkable formula for an integer, when n
  1 we get 4, and n 2 we get 14.

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Counting Sequence Database

• The On-line Encyclopedia of Integer Sequences
  http/www.research.att.com/njas/sequences/index.ht
  ml
• N.J.A. Sloane S. Plouffe, The Encyclopedia of
  Integer Sequences, Academic Press, 1995.

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RNA EQUATIONS

• Recurrence Relations

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RNA EQUATIONS (cont.)

• Generating Function
• 1,1,1,2,4,8,17,37,82,185,423,978,

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RNA EQUATIONS (cont.)

• Exact Formula

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RNA EQUATIONS (cont.)

• s(n,k) is the number of structures of length n
  with exactly k base pairs For n,k gt 0,

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RNA EQUATIONS (cont.)

• Asymptotic Estimate As n grows without bound

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Random Walk

• A random walk is a lattice path from one point to
  another such that steps are allowed in a discrete
  number of directions and are of a certain length

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RNA Walk Type I

• NSE Walks Unit step walks starting at the
  origin (0,0) with steps up, down, and right
• No walks pass below the x-axis and there are no
  consecutive NS steps

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RNA Walk Type I (cont.)

• N (0,1) up
• S (0,-1) down
• E (1,0) right

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Type I Walk Array (n x k)
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RNA Walk Type II

• NSE Walks Unit-step walks starting at the
  origin (0,0) with steps up, down, and right such
  that no walks pass below the x-axis and there are
  no consecutive SN steps

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Type II Walk Array (n x k)
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Examples

• Type I ENNESNESSE
• Type II NEEENSEEES

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RNA Walk Bijection

• Theorem There is a bijection between the set of
  NSE walks of length n1 ending at height k 0
  and the set of NSE walks of length n ending at
  height k 0.
• Source Lattice paths, generating functions, and
  the Riordan group, Ph.D. Thesis, Howard
  University, Washington, DC, 1997

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Matrices Count Lattice Walks

• Type I Walks
• 1 0 0 0 0 0 0 -
• 1 1 0 0 0 0 0 -
• 1 2 1 0 0 0 0 -
• 2 3 3 1 0 0 0 -
• 4 6 6 4 1 0 0 -
• 8 13 13 10 5 1 0 -
• 17 28 30 24 15 6 1 -
• - - - - - - - -

• Type II Walks
• 1 0 0 0 0 0 0 -
• 1 1 0 0 0 0 0 -
• 2 2 1 0 0 0 0 -
• 4 4 3 1 0 0 0 -
• 8 9 7 4 1 0 0 -
• 17 20 17 11 5 1 0 -
• 37 41 41 29 16 6 1 -
• - - - - - - - -

The ith-jth entry corresponds to the number of
random walks of length i and ending height j.
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Type I Formation Rule (Recurrence)
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The Connection Between RNA and the Walks

• Theorem There is a bijection between the set of
  RNA secondary structures of length n and the set
  of NSE walks ending at height k0.
• Source Lattice paths and RNA secondary
  structures, DIMAC Series in Discrete Math.
  Theoretical Computer Science 34 (1997) 137-147.
  (CAARMS2 Proceedings)

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HIV-1 RNA Sequence Prediction

• We want to construct a lattice walk method to
  predict secondary RNA sequences that code for
  regions of the SL2 and SL3 domains within the
  HIV-1 5 UTR RNA molecule.
• These domains are important for HIV genomic
  packaging

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HIV-1 RNA Structural Components
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Components of Secondary Structure

• Base pairs
• Bulges
• Interior Loops
• End loops
• Hairpin
• Multibranch loops junctions where more than one
  hairpin or more complex secondary structures are
  appended.

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HIV-1 Sequence (SL2 SL3)

• The following sequence was obtained from the NCBI
  website. The first 363 nucleotides were
  extracted from the entire HIV-1 RNA genomic
  sequence
• GGUCUCUCUGGUUAGACCAGAUCUGAGCCUGGGAGCUCUCUGGCUAACUA
  GGGAACCCACUGCUUAAGCCUCAAUAAAGCUUGCCUUGAGUGCUUCAAGU
  AGUGUGUGCCCGUCUGUUGUGUGACUCUGGUAACUAGAGAUCCCUCAGAC
  CCUUUUAGUCAGUGUGGAAAAUCUCUAGCAGUGGCGCCCGAACAGGGACC
  UGAAAGCGAAAGGGAAACCAGAGGAGCUCUCUCGACGCAGGACUCGGCUU
  GCUGAAGCGCGCACGGCAAGAGGCGAGGGGCGGCGACUGGUGAGUACGCC
  AAAAAUUUUGACUAGCGGAGGCUAGAAGGAGAGAGAUGGGUGCGAGAGCG
  UCAGUAUUAAGCG
• Color key
• SL2 yellow
• SL3 - red

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Known Sequence of the SL2 Domain
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Lattice Walk Model

• Start with an RNA primary sequence
• Perform RNA combinatorial analysis on the given
  sequence
• Connect lattice walks to the given sequence using
  Type I and II walks
• Calculate identified sequences to find the
  minimum free energy
• Predict secondary sequence
• Conduct laboratory experiments for biological
  functionality

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Acknowledgments

• National Science Foundation, DIMACS, AIMS,
  Burroughs Wellcome, SACEMA, WITS
• MATH. Modeling 561, Graduate Students
• Collaborators Dwayne Hill, Biology Dept., MSU,
  and Alvin Kennedy, Chemistry Dept., MSU