On the power of parallel communicating Watson–Crick automata systems

AbstractParallel communicating Watson–Crick automata systems were introduced in [E. Czeizler, E. Czeizler, Parallel communicating Watson–Crick automata systems, in: Z. Ésik, Z. Fülöp (Eds.), Proc. Automata and Formal Languages, Dobogókő, Hungary, 2005, pp. 83–96] as possible models of DNA computations. This combination of Watson–Crick automata and parallel communicating systems comes as a natural extension due to the new developments in DNA manipulation techniques. It is already known, see [D. Kuske, P. Weigel, The Role of the Complementarity Relation in Watson–Crick Automata and Sticker Systems, DLT 2004, Lecture Notes in Computer Science, Vol. 3340, Auckland, New Zealand, 2004, pp. 272–283], that for Watson–Crick finite automata, the complementarity relation plays no active role. However, this is not the case when considering parallel communicating Watson–Crick automata systems. In this paper we prove that non-injective complementarity relations increase the accepting power of these systems. We also prove that although Watson–Crick automata are equivalent to two-head finite automata, this equivalence is not preserved when comparing parallel communicating Watson–Crick automata systems and multi-head finite automata

On the size of the inverse neighborhoods for one-dimensional reversible cellular automata

AbstractIn this paper we investigate the possible neighborhood size of the inverse automaton of some types of one-dimensional reversible cellular automata. Considering only the case when the local function is a size two map, we give a quadratic upper bound for the neighborhood size of the inverse automaton. We show that this bound can be lowered in some particular cases, and give an algorithm for computing these better bounds

Quantitative Model Refinement as a Solution to the Combinatorial Size Explosion of Biomodels

AbstractBuilding a large system through a systematic, step-by-step refinement of an initial abstract specification is a well established technique in software engineering, not yet much explored in systems biology. In the case of systems biology, one starts from an abstract, high-level model of a biological system and aims to add more and more details about its reactants and/or reactions, through a number of consecutive refinement steps. The refinement should be done in a quantitatively correct way, so that (some of) the numerical properties of the model (such as the experimental fit and validation) are preserved. In this study, we focus on the data-refinement mechanism where the aim is to increase the level of details of some of the reactants of a given model. That is, we analyse the case when a model is refined by substituting a given species by several types of subspecies. We show in this paper how the refined model can be systematically obtained from the original one. As a case study for this methodology we choose a recently introduced model for the eukaryotic heat shock response, [I. Petre, A. Mizera, C. L. Hyder, A. Meinander, A. Mikhailov, R.I. Morimoto, L. Sistonen, J. E. Eriksson, R.-J. Back, A simple mass-action model for the eukaryotic heat shock response and its mathematical validation, Natural Computing, 10(1), 595–612, 2011.]. We refine this model by including details about the acetylation of the heat shock factors and its influence on the heat shock response. The refined model has a significantly higher number of kinetic parameters and variables. However, we show that our methodology allows us to preserve the experimental fit/validation of the model with minimal computational effort

An extension of the Lyndon–Schützenberger result to pseudoperiodic words

AbstractOne 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 θ(u). Thus, any expression consisting of repetitions of u and θ(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form ul=vnwm, 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 θ(t). Moreover, if l⩾5, then n=m=3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement θ(u), which is also obtained in this paper

A tight linear bound on the synchronization delay of bijective automata

AbstractReversible cellular automata (RCA) are models of massively parallel computation that preserve information. We generalize these systems by introducing the class of ωωbijective finite automata. It consists of those finite automata where for any bi-infinite word there exists a unique path labelled by that word. These systems are strictly included in the class of local automata. Although the synchronization delay of an n-state local automaton is known to be Θ(n2) in the worst case, we prove that in the case of ωωbijective finite automata the synchronization delay is at most n−1. Based on this we prove that for a one-dimensional n-state RCA where the neighborhood consists of m consecutive cells, the neighbourhood of the inverse automaton consists of at most nm−1−(m−1) cells. Similar bounds are obtained also in [E. Czeizler, J. Kari, A tight linear bound on the neighborhood of inverse cellular automata, in: Proceedings of ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 410–420] but here the result comes as a direct consequence of the more general result. We also construct examples of RCA with large inverse neighbourhoods proving that the upper bounds provided here are the best possible in the case m=2

Geometrical Tile Design for Complex Neighborhoods

Recent research has showed that tile systems are one of the most suitable theoretical frameworks for the spatial study and modeling of self-assembly processes, such as the formation of DNA and protein oligomeric structures. A Wang tile is a unit square, with glues on its edges, attaching to other tiles and forming larger and larger structures. Although quite intuitive, the idea of glues placed on the edges of a tile is not always natural for simulating the interactions occurring in some real systems. For example, when considering protein self-assembly, the shape of a protein is the main determinant of its functions and its interactions with other proteins. Our goal is to use geometric tiles, i.e., square tiles with geometrical protrusions on their edges, for simulating tiled paths (zippers) with complex neighborhoods, by ribbons of geometric tiles with simple, local neighborhoods. This paper is a step toward solving the general case of an arbitrary neighborhood, by proposing geometric tile designs that solve the case of a “tall” von Neumann neighborhood, the case of the f-shaped neighborhood, and the case of a 3 × 5 “filled” rectangular neighborhood. The techniques can be combined and generalized to solve the problem in the case of any neighborhood, centered at the tile of reference, and included in a 3 × (2k + 1) rectangle

The Complexity of Fixed-Height Patterned Tile Self-Assembly

We characterize the complexity of the PATS problem for patterns of fixed height and color count in variants of the model where seed glues are either chosen or fixed and identical (so-called non-uniform and uniform variants). We prove that both variants are NP-complete for patterns of height 2 or more and admit O(n)-time algorithms for patterns of height 1. We also prove that if the height and number of colors in the pattern is fixed, the non-uniform variant admits a O(n)-time algorithm while the uniform variant remains NP-complete. The NP-completeness results use a new reduction from a constrained version of a problem on finite state transducers.Comment: An abstract version appears in the proceedings of CIAA 201

Generalised Lyndon-Schützenberger Equations

We fully characterise the solutions of the generalised Lyndon-Schützenberger word equations $u_1 \cdots u_\ell = v_1 cdots v_m w_1 \cdots w_n$, where $u_i \in \{u, \theta(u)\}$ for all $1 \leq i \leq \ell$, $v_j \in \{v, \theta(v)\}$ for all $1 \leq j \leq m$, $w_k \in \{w, \theta(w)\}$ for all $1 \leq k ?\leq n$, and $\theta$ is an antimorphic involution. More precisely, we show for which $\ell$, $m$, and $n$ such an equation has only $\theta$-periodic solutions, i.e., $u$, $v$, and $w$ are in $\{t, \theta(t)\}^\ast$ for some word $t$, closing an open problem by Czeizler et al. (2011)

Search methods for tile sets in patterned DNA self-assembly

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