We examine scattered hairpins, which are structures formed when a single strand folds into a partially hybridized stem and a loop To specify different classes of hairpins, we use the concept of DNA trajectories, which allows precise descriptions of valid bonding patterns on the stem of the hairpin DNA trajectories have previously been used to describe bonding between separate strands

We are interested in the mathematical properties of scattered hairpins described by DNA trajectories We examine the complexity of set of hairpin-free words described by a set of DNA trajectories In particular, we consider the closure properties of language classes under sets of DNA trajectories of differing complexity We address decidability of recognition problems for hairpin structures

background

/pdf/hairpin-structures-defined-by-dna-trajectories-3d4geq0ayf.pdf

Generating DNA Code Words.

Hairpin structures defined by DNA trajectories

We examine scattered hairpins, which are structures formed when a single strand of nucleotides folds into a partially hybridized stem and a loop. To specify different classes of hairpins, we use the concept of DNA trajectories, which allows precise descriptions of valid bonding patterns on the stem of the hairpin. DNA trajectories have previously been used to describe bonding between separate strands.
 
We are interested in the mathematical properties of scattered hairpins described by DNA trajectories. We examine the complexity of the set of hairpin-free words described by a set of DNA trajectories. In particular, we consider the closure properties of language classes under sets of DNA trajectories of differing complexity. We address decidability of recognition problems for hairpin structures.

/pdf/hairpin-structures-defined-by-dna-trajectories-2l2wvkk4yb.pdf

Hairpin Structures Defined by DNA Trajectories

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.

Preventing undesirable bonds between DNA codewords

The concepts of pseudo-bordered and pseudo-unbordered words are in large part motivated by research in theoretical DNA computing, wherein the Watson---Crick complementarity of DNA strands is modelled as an antimorphic involution, that is, a function $$\theta $$ź which is an antimorphism, $$\theta (uv)=\theta (v) \theta (u)$$ź(uv)=ź(v)ź(u), and an involution, $$\theta (\theta (u))=u$$ź(ź(u))=u, for all words u, v over the DNA alphabet. In particular, a word w is said to be $$\theta $$ź-bordered (or pseudo-bordered) if there exists a word $$v \in \Sigma ^{+}$$vźΣ+ that is a proper prefix of w, while $$\theta (v)$$ź(v) is a proper suffix of w. A word which is not $$\theta $$ź-bordered is $$\theta $$ź-unbordered. This paper continues the exploration of properties (for the case where $$\theta $$ź is a morphic involution) of the set of $$\theta $$ź-unbordered words, $$D_{\theta }(1)$$Dź(1), and the sets of words that have exactly i$$\theta $$ź-borders, $$D_{\theta }(i)$$Dź(i), $$i \ge 2$$iź2. We prove that, under some conditions, the set $$D_{\theta }(i)$$Dź(i) is disjunctive for all $$i \ge 1$$iź1, and that the set $$D_{\theta }^i(1){\setminus }D(i)$$Dźi(1)\D(i) is disjunctive for all $$i\ge 2$$iź2, where D(i) denotes the set of words with exactly i borders. We also discuss conditions for catenations of languages of $$\theta $$ź-unbordered words to remain $$\theta $$ź-unbordered, and anticipate further generalizations by showing that the set of all $$\theta $$ź-bordered words is not context-free for all morphisms $$\theta $$ź over an alphabet $$\Sigma $$Σ with $$|\Sigma | \ge 3$$|Σ|ź3 such that $$\theta (a) \ne a$$ź(a)źa for all $$a \in \Sigma $$aźΣ and $$\theta ^{3}$$ź3 equals the identity function on $$\Sigma $$Σ.

Disjunctivity and other properties of sets of pseudo-bordered words

The set of all sequences that are generated by a biomolecular protocol forms a language over the four letter alphabet Δ={A,G,C,T}. This alphabet is associated with a natural involution mapping θ, A↦T and G↦C which is an antimorphism of Δ*. In order to avoid undesirable Watson-Crick bonds between the words (undesirable hybridization), the language has to satisfy certain coding properties. In this paper we build upon an earlier initiated study and give general methods for obtaining sets of code words with the same properties. We show that some of these code words have enough entropy to encode {0,1}* in a symbol-to-symbol mapping.

Generating DNA Code Words.

Citations

Hairpin structures defined by DNA trajectories

Hairpin Structures Defined by DNA Trajectories

Preventing undesirable bonds between DNA codewords

Disjunctivity and other properties of sets of pseudo-bordered words

Methods for Constructing Coded DNA Languages

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