LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

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Methods for Constructing Coded DNA Languages 245
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u = k
Fig. 3. Various cross hybridizations of molecules one of which contains subword of
length k and the other its complement.
7. If X is a θ-k-code then X and θ(X) avoidsallcrosshybridizationsoflength
k shown in Fig. 3 and so all cross hybridizations presented in Fig. 2 of [16].
It is clear that θ-subword compliance implies θ-prefix and θ-suffix compliance.
We note that when θ = ρν, theθ(k, m1,m2)-subword compliance of the code
words X ⊆ ∆ ∗ does not allow intramolecular hybridization as in Fig. 1 for a predetermined
k and m1 ≤ m ≤ m2. The maximal length of a word that together
with its reverse complement can appear as subwords of code words is limited
with k. The length of the hairpin, i.e. “distance” between the word and its reversed
complement is bounded between m1 and m2. Thevaluesofk and m1,m2
would depend on the laboratory conditions (ex. the melting temperature, the
length of the code words and the particular application). In order to avoid intermolecular
hybridizations as presented in Fig. 2, X has to satisfy θ-compliance
and θ-freedom. Most applications would require X to be strictly θ. Themost
restricted and valuable properties are obtained with θ-k-code, and the analysis
of this type of codes also seems to be more difficult. When X is θ-k-code, all
intermolecular hybridizations presented in Fig. 3 are avoided. We include several
observations in the next section.
2.1 Closure Properties
In this part of the paper we consider several closure properties of languages that
satisfy some of the properties described with the Definition 21. We concentrate
on closure properties of union and concatenation of “good” code words. From
practical point of view, we would like to know under what conditions two sets of
available “good” code words can be joined (union) such that the resulting set is
still a “good” set of code words. Also, whenever ligation of strands is involved,
we need to consider concatenation of code words. In this case it is useful to
know under what conditions the “good” properties of the codewords will be
preserved after arbitrary ligations. The following table shows closure properties
of these languages under various operations.
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