LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Methods for Constructing Coded DNA Languages 251
Let A be the adjacency matrix of this automaton with the characteristic
equation µ 2k−2 − µ k−1 (p − 1) k−2 − (p − 1) k−2 =0.Thisimplies(p− 1) k−2 =
µ k−1 − µk−1
µ k−1 . We are interested in the largest real value for µ. Sinceµ>0, we
+1
have 0 < µk−1
µ k−1 +1
< 1 which implies (p − 1) k−2
k−1 3 2
3 > 2. Hence the entropy of X ∗ in this case is greater than log 2.
4 Concluding Remarks
In this paper we investigated theoretical properties of languages that consist of
DNA based code words. In particular we concentrated on intermolecular and
intramolecular cross hybridizations that can occur as a result that a Watson-
Crick complement of a (sub)word of a code word is also a (sub)word of a code
word. These conditions are necessary for a design of good codes, but certainly
may not be sufficient. For example, the algorithms used in the programs developed
by Seeman [23], Feldkamp [10] and Ruben [22], all check for uniqueness of
k-length subsequences in the code words. Unfortunately, none of the properties
from Definition 21 ensures uniqueness of k-length words. Such code word properties
remain to be investigated. We hope that the general methods of designing
such codewords will simplify the search for “good” codes. Better characterizations
of good code words that are closed under Kleene ∗ operation may provide
even faster ways for designing such codewords. Although the Proposition 36
provides a rather good design of code words, the potential repetition of certain
subwords is not desirable. The most challenging questions of characterizing and
designing good θ-k-codes that avoids numerous repetition of subwords remains
to be developed.
Our approach to the question of designing “good” DNA codes has been from
the formal language theory aspect. Many issues that are involved in designing
such codes have not been considered. These include (and are not limited to)
the free energy conditions, melting temperature as well as Hamming distance
conditions. All these remain to be challenging problems and a procedure that
includes all or majority of these aspects will be desirable in practice. It may be
the case that regardless of the way the codes are designed, the ultimate test for
the “goodness” of the codes will be in the laboratory.
Acknowledgment
This work has been partially supported by the grant EIA-0086015 from the
National Science Foundation, USA.
References
1. R.L. Adler, D. Coppersmith, M. Hassner, Algorithms for sliding block codes -an application
of symbolic dynamics to information theory, IEEE Trans. Inform. Theory
29 (1983), 5-22.