Algorithmes de prediction et de recherche de multi-structures d'ARN

26 Chapter 2. Defining RNA structures on base pairs and on helices
Figure 2.1: Secondary structure of transfer RNA (tRNA) Ala, seen as a set of base pairs (top) or as a set of
helices (bottom). Each rectangle shows a group of adjacent bases included in a helix. Between two rectangles
belonging to the same helix a gras edge is appeared.
A secondary structure on BP is a subset of BP such that any two distinct base pairs of S are
either nested or juxtaposed. Figure 2.2 gives an example of a secondary structure with 6 base
pairs.
In the rest of this thesis, a secondary structure is decomposed into nested levels of juxtaposed
base pairs. For this perspective, we define the two following notations:
• Given a structure S, toplevel(S) is defined as the set of base pairs of S that are not
nested in any base pair of S.
• Given a base pair (x, y) in S, nested(x, y, S) is the set of base pairs of S that are nested
in (x, y), and that are not nested in any base pair nested in (x, y).
These levels induce a partition of S:
S = toplevel(S) ∪ �
(x,y)∈S nested(x, y, S)
For example, on the Figure 2.2, toplevel(S) = {(1, 6), (5, 10), (9, 14)}, nested(1, 6, S) =
{(2, 4), (3, 5)}, nested(5, 10, S) = {(7, 8)}, nested(9, 14, S) = {(10, 11), (12, 13)}, and all other
nested sets are empty.
It is routine to verify that S is a secondary structure if, and only if, any two base pairs of
toplevel(S) are juxtaposed, and for each (x, y) of S, any two base pairs of nested(x, y, S) are
juxtaposed.
Finally, we also define intervals which are subsets of the base pair set. Given a set BP of base
pairs, BP[x..y] denotes the subset of BP composed of base pairs (z, t) such that x ≤ z < t ≤ y.