Extensions of rich words

In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The unified study of words which achieve this limit was initiated in [A. Glen et al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these words rich (in palindromes). This article contains several results about rich words and especially extending them. We say that a rich word w can be extended richly with a word u if wu is rich. Some notions are also made about the infinite defect of a word, the number of rich words of length n and two-dimensional rich words.Comment: 19 pages, 3 figure

Languages invariant under more symmetries: overlapping factors versus palindromic richness

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur

Phylogenetic differences in content and intensity of periodic proteins

Many proteins exhibit sequence periodicity, often correlated with a visible structural periodicity. The statistical significance of such periodicity can be assessed by means of a chi-square-based test, with significance thresholds being calculated from shuffled sequences. Comparison of the complete proteomes of 45 species reveals striking differences in the proportion of periodic proteins and the intensity of the most significant periodicities. Eukaryotes tend to have a higher proportion of periodic proteins than eubacteria, which in turn tend to have more than archaea. The intensity of periodicity in the most periodic proteins is also greatest in eukaryotes. By contrast, the relatively small group of periodic proteins in archaea also tend to be weakly periodic compared to those of eukaryotes and eubacteria. Exceptions to this general rule are found in those prokaryotes with multicellular life-cycle phases, e.g. Methanosarcina sps. or Anabaena sps., which have more periodicities than prokaryotes in general, and in unicellular eukaryotes, which have fewer than multicellular eukaryotes. The distribution of significantly periodic proteins in eukaryotes is over a wide range of period lengths, whereas prokaryotic proteins typically have a more limited set of period lengths. This is further investigated by repeating the analysis on the NRL-3D database of proteins of solved structure. Some short range periodicities are explicable in terms of basic secondary structure, e.g. alpha helices, while middle range periodicities are frequently found to consist of known short Pfam domains, e.g. leucine-rich repeats, tetratricopeptides or armadillo domains. However, not all can be explained in this way

Scale-Free topologies and Activatory-Inhibitory interactions

A simple model of activatory-inhibitory interactions controlling the activity of agents (substrates) through a "saturated response" dynamical rule in a scale-free network is thoroughly studied. After discussing the most remarkable dynamical features of the model, namely fragmentation and multistability, we present a characterization of the temporal (periodic and chaotic) fluctuations of the quasi-stasis asymptotic states of network activity. The double (both structural and dynamical) source of entangled complexity of the system temporal fluctuations, as an important partial aspect of the Correlation Structure-Function problem, is further discussed to the light of the numerical results, with a view on potential applications of these general results.Comment: Revtex style, 12 pages and 12 figures. Enlarged manuscript with major revision and new results incorporated. To appear in Chaos (2006

Michaelis-Menten Dynamics in Complex Heterogeneous Networks

Biological networks have been recently found to exhibit many topological properties of the so-called complex networks. It has been reported that they are, in general, both highly skewed and directed. In this paper, we report on the dynamics of a Michaelis-Menten like model when the topological features of the underlying network resemble those of real biological networks. Specifically, instead of using a random graph topology, we deal with a complex heterogeneous network characterized by a power-law degree distribution coupled to a continuous dynamics for each network's component. The dynamics of the model is very rich and stationary, periodic and chaotic states are observed upon variation of the model's parameters. We characterize these states numerically and report on several quantities such as the system's phase diagram and size distributions of clusters of stationary, periodic and chaotic nodes. The results are discussed in view of recent debate about the ubiquity of complex networks in nature and on the basis of several biological processes that can be well described by the dynamics studied.Comment: Paper enlarged and modified, including the title. Some problems with the pdf were detected in the past. If they persist, please ask for the pdf by e-mailing yamir(at_no_spam)unizar.es. Version to appear in Physica

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word