On properties of bond-free DNA languages

Abstract

The input data for DNA computing must be encoded into the form of single or double DNA strands. As complementary parts of single strands can bind together forming a double-stranded DNA sequence, one has to impose restrictions on these sets of DNA words (languages) to prevent them from interacting in undesirable ways. We recall a list of known properties of DNA languages which are free of certain types of undesirable bonds. Then we introduce a general framework in which we can characterize each of these properties by a solution of a uniform formal language inequation. This characterization allows us among others to construct (i) a uniform algorithm deciding in polynomial time whether a given DNA language possesses any of the studied properties, and (ii) in many cases also an algorithm deciding whether a given DNA language is maximal with respect to the desired property.

Hairpin Structures Defined by DNA Trajectories

“…This is similar to the work on bond-free properties [13] and hairpin-freeness [12,14]. In practice, an antimorphic involution yields results applicable to hairpin and scatteredhairpin structures, while morphic involutions yield structures where the scattered stem is bonded in a parallel, rather than an anti-parallel, orientation.…”

“…We now consider DNA trajectories, defined by Kari et al [13]. A DNA trajectory is a word over the alphabet…”

“…The occurrence of b b implies a bond at a certain position, while f f (resp., f ε , ε f ) denotes two bases which are free (resp., an extra unbonded nucleotide on the top strand, an extra unbonded nucleotide on the bottom strand). DNA trajectories are used to define so-called bond-free properties in DNA code word design [13], and we adopt them here for modelling the bonding of hairpins.…”

Preventing Undesirable Bonds Between DNA Codewords

“…Solid languages are also used in [10] as a tool for constructing error-detecting DNA languages that are invariant under bio-operations. We refer to [6,10,11] for further examples and for mutual relations between classes of a DNA languages satisfying these properties.…”

“…Polynomial-time algorithms are presented for the case of θ-compliant finite sets and θ-non-overlapping regular sets. The proofs of these results are mostly nontrivial and due to page limitations can be found in [11].…”