Shinnosuke Seki

On a special class of primitive words

When representing DNA molecules as words, it is necessary to take into account the fact that a word u encodes basically the same information as its Watson-Crick complement @q(u), where @q denotes the Watson-Crick complementarity function. Thus, an expression which involves only a word u and its complement can be still considered as a repeating sequence. In this context, we define and investigate the properties of a special class of primitive words, called pseudo-primitive words relative to @q or simply @q-primitive words, which cannot be expressed as such repeating sequences. For instance, we prove the existence of a unique @q-primitive root of a given word, and we give some constraints forcing two distinct words to share their @q-primitive root. Also, we present an extension of the well-known Fine and Wilf theorem, for which we give an optimal bound.

CHARACTERIZATIONS OF BOUNDED SEMILINEAR LANGUAGES BY ONE-WAY AND TWO-WAY DETERMINISTIC MACHINES

A bounded language

Scalable, time-responsive, digital, energy-efficient molecular circuits using DNA strand displacement

L \subseteq x_1^* \cdots x_k^*

An Extension of the Lyndon Schützenberger Result to Pseudoperiodic Words

(for some k ≥ 1 and not-necessarily distinct nonempty words x1, …, xk) is bounded semilinear if the set

K-Comma Codes and Their Generalizations

Q(L) =\{(i_1, \ldots, i_k)\, |\, x_1^{i_1} \cdots x_k^{i_k} \in L\}

Combinatorial Optimization in Pattern Assembly

is semilinear. We give characterizations of bounded semilinear languages in terms of one-way and two-way deterministic counter machines.

On pseudoknot-bordered words and their properties

We propose a novel theoretical biomolecular design to implement any Boolean circuit using the mechanism of DNA strand displacement. The design is scalable: all species of DNA strands can in principle be mixed and prepared in a single test tube, rather than requiring separate purification of each species, which is a barrier to large-scale synthesis. The design is time-responsive: the concentration of output species changes in response to the concentration of input species, so that time-varying inputs may be continuously processed. The design is digital: Boolean values of wires in the circuit are represented as high or low concentrations of certain species, and we show how to construct a single-input, single-output signal restoration gate that amplifies the difference between high and low, which can be distributed to each wire in the circuit to overcome signal degradation. This means we can achieve a digital abstraction of the analog values of concentrations. Finally, the design is energy-efficient: if input species are specified ideally (meaning absolutely 0 concentration of unwanted species), then output species converge to their ideal concentrations at steady-state, and the system at steady-state is in (dynamic) equilibrium, meaning that no energy is consumed by irreversible reactions until the input again changes.

Binary Pattern Tile Set Synthesis Is NP-Hard

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson-Crick complement, denoted here as ?(u). Thus, any expression consisting of repetitions of u and ?(u) can be considered in some sense periodic. In this paper we give a generalization of Lyndon and Schutzenbergers classical result about equations of the form u l = v n w m , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l ? 5, n, m ? 3, then all three words involved can be expressed in terms of a common word t and its complement ?(t). Moreover, if l ? 5, then n = m = 3 is an optimal bound. We also obtain a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement ?(u).

Programming Biomolecules That Fold Greedily During Transcription

In this paper, we introduce the notion of k-comma codes - a proper generalization of the notion of comma-free codes. For a given positive integer k, a k-comma code is a set L over an alphabet Σ with the property that LΣ kL ∩ Σ +LΣ + = ∅. Informally, in a k-comma code, no codeword can be a subword of the catenation of two other codewords separated by a “comma” of length k. A k-comma code is indeed a code, that is, any sequence of codewords is uniquely decipherable. We extend this notion to that of k-spacer codes, with commas of length less than or equal to a given k. We obtain several basic properties of k-comma codes and their generalizations, k-comma intercodes, and some relationships between the families of k-comma intercodes and other classical families of codes, such as infix codes and bifix codes. Moreover, we introduce the notion of n-k-comma intercodes, and obtain, for each k ≥ 0, several hierarchical relationships among the families of n-k-comma intercodes, as well as a characterization of the family of 1-k-comma intercodes.

On a Special Class of Primitive Words

Pattern self-assembly tile set synthesis (Pats) is an NP-hard combinatorial problem to minimize a rectilinear tile assembly system (RTAS) that uniquely self-assembles a given rectangular pattern. For c ≥ 1, c-Pats is a subproblem of Pats which takes only the patterns with at most c colors as input. We propose a polynomial-time reduction of 3Sat to 60-Pats in order to prove that 60-Pats is NP-hard.