LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

An Algorithm for Testing Structure Freeness of Biomolecular Sequences 271
ii +1
···•=⇒•=⇒•··· •
| ⇓
···•⇐=•⇐=•··· •
jj − 1
(a) Hairpin
ii +1 p − 1p
···•=⇒•=⇒•··· •=⇒•=⇒•···
| |
···•⇐=•⇐=•··· •⇐=•⇐=•···
jj − 1 q +1q
1 i − 1i
•=⇒···=⇒•=⇒•···
|
•⇐=···⇐=•⇐=•···
N j +1j
(b) Free End
(c) Internal Loop
(Bulge Loop in case of i +1=p or j = q +1)
(Stacked Pair in case of i +1=p and j = q +1)
Fig. 1. Basic Secondary Structures
β,γ ∈ S. Then, (i, j) is said to have configuration (β,k,γ,l) inα(T )iftheith
and jth bases of α correspond to the kth base of the segment β and the lth base
of the segment γ, respectively. More formally, (i, j) issaidtohave configuration
(β,k,γ,l) in α(T )ifthereexistx, y, z, w ∈ S∗ such that α = xβy = zγw,
|x| bpm, is defined as a sequence cf(bp1), ..., cf(bpm). For two structured
strings α1(T1) andα2(T2), we write α1(T1) ≡ α2(T2)ifcf(α1(T1)) = cf(α2(T2)).
A structured string α(T )issaidtobeE-minimal if for any α ′ (T ′ )such
that α(T ) ≡ α ′ (T ′ ), E(α(T )) ≤ E(α ′ (T ′ )) holds. The existence of such an Eminimal
structured string is not clear at this point. But, we can show that for
any configuration C, there always exists an E-minimal structured string α(T )
such that cf(α(T )) = C, which will be implicitly proved in the proof of (2)→(1)
in Theorem 2.
A2-cyclec with base pairs bp1,bp2 (bp2 < bp1) inα(T )issaidtohave
boundary configuration (v1,v2) ifcf(bpi) =vi (i =1, 2). A 1-cycle or a free end
structure c with a base pair bp in α(T ) is said to have the boundary configuration
v if cf(bp) =v.