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

42 Chapter 3. Regliss – Locally optimal structures prediction
(a) Initial set of base pairs
(b) Locally optimal secondary structures
1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 4 6 7 8 9 10 11 12 13 14 1 3 5 6 7 8 9 10 11 12 13 14 2 4 5 7 8 10 12 13
(c) All possible trees corresponding to the construction of locally optimal structures from structures
maximal for juxtaposition
MJ(2, 5)[0] = {(2, 4)}
MJ(3, 3)[0] = ε
MJ(1, 14)[0] = {(1, 6), (7, 8), (9, 14)}
MJ(8, 7)[0] = ε MJ(10, 13)[0] = {(10, 11), (12, 13)}
MJ(11, 10)[0] = ε
(d) States of the push-down stack
(1, 14, −1)
initialization
MJ(13, 12)[0] = ε
MJ(2, 5)[1] = {(3, 5)}
MJ(4, 4)[0] = ε
MJ(1, 14)[1] = {(2, 4), (5, 10), (12, 13)}
MJ(3, 3)[0] = ε MJ(6, 9)[0] = {(7, 8)} MJ(13, 12)[0] = ε
(10, 13, 0)
(2, 5, 0)
(1, 14, 0 )
iteration 1
MJ(8, 7)[0] = ε
(10, 13, 0)
(2, 5, 1 )
(1, 14, 0)
iteration 2
14
MJ(1, 14)[0] = {(1, 6), (7, 8), (9, 14)}
MJ(8, 7)[0] = ε MJ(10, 13)[0] = {(10, 11), (12, 13)}
MJ(11, 10)[0] = ε MJ(13, 12)[0] = ε
(6, 9, 0)
(1, 14, 1 )
iteration 3
Figure 3.7: (a) and (b) come from Figure 3.3. (c) Trees associated to the three locally optimal secondary
structures. If (i, j) is a base pair of BP, then MJ(i, j)[k] denotes the kth element of MJ(i, j). (d) Contents of
the stack at the end of each iteration of the algorithm of Figure 3.8. MJ(i, j)[k] is symbolized by the triplet
(i, j, k). Each iteration outputs a locally optimal structure. Each cell of the stack corresponds to an internal node
of the underlying tree depicted in (c). In each iteration, the boxed element is the new MJ(i, j)[k + 1], pushed
by line 5 of the algorithm. All bold elements are then pushed through initializations of lines 6 and 7: their last
component is always 0. For example, at iteration 1, the line 5 pushes the triplet (1, 14, 0), corresponding to
MJ(1, 14)[0] = {(1, 6), (7, 8), (9, 14)}, then the nested structures are pushed.