LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Remarks on Relativisations and DNA Encodings 137
Now we can define the involution τ = µγ = γµ on ∆ ∗ , modeling the Watson-
Crick mapping between complementary single-stranded DNA molecules.
Results from Section 3, and the fact that J(C) ⊆ C, allow us to restate
Proposition 4 from [2] (quoted in our Section 2) as follows: if C is a code, then
J(C) isI-free. In a similar way, we want a definition of Jτ(C), subset of C, so
that Jτ(C) isτ-free.
The experimental motivation of this goal is that τ-freedom, and not Ifreedom
or relative solidity, would protect from the effects of unwanted reverse
complementarity. I-freedom guarantees that if we put a set of probes from J(C)
in the test tube, they will correctly hybridize to the hypothetical longer DNA
molecule, even if this is built from the larger set C. Nonetheless, among words
in J(C) there could still be words which can hybridize with words from the
same set J(C), interfering with the desired reactions. This would be formally
equivalent to say that J(C) isnotτ-free.
We conjecture that we could define Jτ (C)asbeingrelatedtoJ(C), as follows:
Jτ(C) =J(C)\τ(J(C)).
Such a subset of C would be I-free, since it is a subset of J(C), and would
avoid self hybridization among words of Jτ(C). Further, it would induce a splitting
in each J(C), which would still allow to operate detection, i.e., matching
between probes and segments on a longer molecule, in a relativised, step by step
way, even if with a greater number of steps. Further theoretical studies on this
issue is being carried on.
5 Final Remarks
In order to make these theoretical results actually useful for laboratory DNA
experiments, more work should be done on example codes.
It would be interesting to generate codes which enjoy the properties defined
in Section 4, and to compare them to codes generated as in Proposition 16 of [3]
or as in Proposition 4 of [5].
On the other hand, algorithms could be studied that would decide whether
a given code enjoys properties defined in Section 4, and whether there is a way
of splitting it into a finite number of Jτ (Ck) sub-codes.
References
1. J. Berstel, D. Perrin. Theory of Codes, Academic Press, 1985.
2. T. Head, “Relativised code concepts and multi-tube DNA dictionaries”, submitted,
2002.
3. S. Hussini, L. Kari, S. Konstantinidis. “Coding properties of DNA languages”. Theoretical
Computer Science, 290, pp.1557–1579, 2003.
4. N. Jonoska, D. Kephart, K. Mahaligam. “Generating DNA code words”, Congressum
Numeratium, 156, pp.99-110, 2002.