LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Methods for Constructing Coded DNA Languages 243
For words representing DNA sequences we use the following convention. A
word u over ∆ denotes a DNA strand in its 5 ′ → 3 ′ orientation. The Watson-
Crick complement of the word u, also in orientation 5 ′ → 3 ′ is denoted with
← u. For example if u = AGGC then ← u= GCCT. There are two types of unwanted
hybridizations: intramolecular and intermolecular. The intramolecular
hybridization happens when two sequences, one being a reverse complement of
the other appear within the same DNA strand (see Fig. 1). In this case the DNA
strand forms a hairpin.
u u
v x
(a) (b)
w=uvux, u = k, v = m
Fig. 1. Intramolecular hybridization (θ-subword compliance): (a) the reverse complement
is at the beginning of the 5 ′ end, (b) the reverse complement is at the end of the
3 ′ .The3 ′ endoftheDNAstrandisindicatedwithanarrow.
Two particular intermolecular hybridizations are of interest (see Fig. 2). In
Fig. 2 (a) the strand labeled u is a reverse complement of a subsequence of the
strand labeled v, and in the same figure (b) represents the case when u is the
reverse complement of a portion of a concatenation of v and w.
(a)
u u
v
Fig. 2. Two types of intermolecular hybridization: (a) (θ-compliant) one code word is
a reverse complement of a subword of another code word, (b) (θ-free) a code word is a
reverse complement of a subword of a concatenation of two other code words. The 3 ′
endisindicatedwithanarrow.
Throughout the rest of the paper, we concentrate on finite sets X ⊆ Σ ∗ that
are codes such that every word in X + can be written uniquely as a product
of words in X. Inotherwords,X ∗ is a free monoid generated with X. For
the background on codes we refer the reader to [5]. We will need the following
definitions:
Pref(w) ={u |∃v ∈ Σ ∗ ,uv = w}
Suff(w) ={u |∃v ∈ Σ ∗ ,vu= w}
Sub(w) ={u |∃v1,v2 ∈ Σ ∗ ,v1uv2 = w}
We define the set of prefixes, suffixes and subwords of a set of words. Similarly,
we have Suffk(w) = Suff(w) ∩ Σ k ,Prefk(w) =Pref(w) ∩ Σ k and Subk(w) =
Sub(w) ∩ Σ k .
v
(b)
w