RNA basics: Sequence and Shape Sets Lecture 2A

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RNA basicsSequence and Shape Sets Lecture 2A

• Dr. Eduardo Mendoza
• Mathematics Department Physics
  Department
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph

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Agenda

• Sequences and shapes
• Combinatorics of shape sets
• Connectivity of neutral networks
• Shape space covering

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1. RNA Sequence Set

• Notation
• S(n,A) sequences of length n over the alphabet
  A
• S(n) S(n,A) with A A,U,G,C
• Similarly for shapes Sh(n,A), Sh(n)
• Hamming graph structure (distance) on S(n,A)
  well interpreted as point-wise mutation this is
  the well-known hypercube
• Pairing rule symmetric relation B on alphabet A
• e.g. for A A,U,G,C, pairing rule B (due to
  physical chemical constraints) AU,UA,GC,CG,
  GU,UG.

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Hypercube
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RNA Shapes (secondary structures)

• Definition A secondary structure (or shape) is a
  vertex-labeled (undirected) graph on n vertices
  whose (symmetric) adjacency matrix A (aij)
  has the following properties

HOF94
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Basic terminology for shapes
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Basic structure elements (1)
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Basic structure elements (2)
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(No Transcript)
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BUN01
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Representations of RNA shapes
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Arcs
Rooted Trees
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Neutral Network of a shape

• Let f S(n) ? Sh(n) be the folding map a
  sequence is mapped to the shape with minimum free
  energy (mfe) it folds into.
• Def The neutral network G(s) of a shape s
• is f-1 (s), i.e. all sequences with s as
  mfe-shape.
• A graph structure is induced by Hamming
  distance.
• Biological significance a substantial fraction
  of point mutations has no measurable effect on
  fitness (Neutral theory of evolution, Kimura
  1983)

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Example Neutral Network
SCH01
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Compatible set of a shape

• Pairing rule symmetric relation B on alphabet A
• e.g. for A A,U,G,C, pairing rule B (due to
  physical chemical constraints) AU,UA,GC,CG,
  GU,UG.
• Pairing scheme P(s) of a shape s
• Set of compatible sequences C(s) of a shape s
  x in S(n)/ xi, xj in B iff i,j in P(s)
• Clearly neutral network G(s) is contained in
  C(s).

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2. Combinatorics of RNA Shapes
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Excerpts from CLBA00

• Basic recursion formula (Waterman, Thm. 6.2)
• Exponential growth of shapes (Waterman, Prop.
  6.3)
• Asymptotic Behavior (Stein-Waterman, Rem. P. 205)

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SCH01
Vienna school preprints at http//www.tbi.univi
e.ac.at
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2.2 Common rare shapes
Def
SCH01
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3. Connectivity of neutral networks

• Approach model the neutral network as a random
  graph (initiated by Vienna school 1995)
• Central concept average degree of neutrality
• Percolation Phenomenon a critical threshold
  value exists such that the neutral network
• is partitioned into components with one
  dominating giant part and many small islands
  if average degree of neutrality is below that
  value
• is connected and dense in S(n) if average degree
  of neutrality exceeds the value

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4. Shape Space Covering
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Shape Space Covering (2)
SCH01
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Shape Space Covering (3)
P. Stadler Remark follows from Intersection
Theorem (but how?)
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Possible RNA/Bio-Characters topics

• RNA Interference by G. Hannon, Nature July 11,
  2002 (only Biology, no math/comp science)