Diego Luis Gonzalez

Circular codes, symmetries and transformations

Circular codes, putative remnants of primeval comma-free codes, have gained considerable attention in the last years. In fact they represent a second kind of genetic code potentially involved in detecting and maintaining the normal reading frame in protein coding sequences. The discovering of an universal code across species suggested many theoretical and experimental questions. However, there is a key aspect that relates circular codes to symmetries and transformations that remains to a large extent unexplored. In this article we aim at addressing the issue by studying the symmetries and transformations that connect different circular codes. The main result is that the class of 216

Dinucleotide circular codes and bijective transformations.

C^3

DNA, dichotomic classes and frame synchronization: a quasi-crystal framework.

C3 maximal self-complementary codes can be partitioned into 27 equivalence classes defined by a particular set of transformations. We show that such transformations can be put in a group theoretic framework with an intuitive geometric interpretation. More general mathematical results about symmetry transformations which are valid for any kind of circular codes are also presented. Our results pave the way to the study of the biological consequences of the mathematical structure behind circular codes and contribute to shed light on the evolutionary steps that led to the observed symmetries of present codes.

Detecting structure in parity binary sequences

In 1996 Arquès and Michel [1996. A complementary circular code in the protein coding genes. J. Theor. Biol. 182, 45-58] discovered the existence of a common circular code in eukaryote and prokaryote genomes. Since then, circular code theory has provoked great interest and underwent a rapid development. In this paper we discuss some theoretical issues related to the synchronization properties of coding sequences and circular codes with particular emphasis on the problem of retrieval and maintenance of the reading frame. Motivated by the theoretical discussion, we adopt a rigorous statistical approach in order to try to answer different questions. First, we investigate the covering capability of the whole class of 216 self-complementary, C(3) maximal codes with respect to a large set of coding sequences. The results indicate that, on average, the code proposed by Arquès and Michel has the best covering capability but, still, there exists a great variability among sequences. Second, we focus on such code and explore the role played by the proportion of the bases by means of a hierarchy of permutation tests. The results show the existence of a sort of optimization mechanism such that coding sequences are tailored as to maximize or minimize the coverage of circular codes on specific reading frames. Such optimization clearly relates the function of circular codes with reading frame synchronization.

Three-frequency resonances in coupled phase-locked loops

The presence of circular codes in mRNA coding sequences is postulated to be involved in informational mechanisms aimed at detecting and maintaining the normal reading frame during protein synthesis. Most of the recent research is focused on trinucleotide circular codes. However, also dinucleotide circular codes are important since dinucleotides are ubiquitous in genomes and associated to important biological functions. In this work we adopt the group theoretic approach used for trinucleotide codes in Fimmel et al. (2015) to study dinucleotide circular codes and highlight their symmetry properties. Moreover, we characterize such codes in terms of n-circularity and provide a graph representation that allows to visualize them geometrically. The results establish a theoretical framework for the study of the biological implications of dinucleotide circular codes in genomic sequences.

One-dimensional poincare map for a non-linear driven oscillator: Analytical derivation and geometrical properties☆

We investigate the hierarchical structure of three-frequency resonances in nonlinear dynamical systems with three interacting frequencies. We hypothesize an ordering of these resonances based on a generalization of the Farey tree organization from two frequencies to three. In experiments and numerical simulations we demonstrate that our hypothesis describes the hierarchies of three-frequency resonances in representative dynamical systems. We conjecture that this organization may be universal across a large class of three-frequency systems. @S1063-651X~99!14803-7# PACS number~s!: 05.45.2a Nonlinear systems with two competing frequencies show resonances or lockings, in which the system locks into a resonant periodic response which has a rational frequency ratio @1#. The locking increases with nonlinearity, from none in the linear regime, to a critical situation where the system is everywhere resonant. The subcritical system has quasiperiodic responses between different lockings, while at supercritical values of the nonlinearity, chaotic as well as periodic and quasiperiodic responses may occur. Resonances have been investigated theoretically and experimentally in many nonlinear systems, and their distribution in parameter space in the form of a devil’s staircase is now well understood, from the number theoretical concept of Farey trees @2‐7#. However, all this applies to resonances generated by the interaction of two frequencies. Far less is known, by comparison, when there are three or more interacting frequencies. Adding another frequency allows new phenomena to take place. Now as well as ~two-frequency! resonance as before, there is a further possibility: three-frequency resonance, also known as weak resonance or partial mode locking. Threefrequency resonances are given by the nontrivial solutions of the equation af 0 1 bf 1 1 cf 2 50, where a, b, and c are integers, f 1 and f 2 are the forcing frequencies, and f 0 is the resonant response. They form a web in the parameter space of the frequencies @8‐11#. In this paper, we hypothesize a local ordering of three-frequency resonances based on generalizing the Farey tree of two-frequency systems to three frequencies. We perform experiments and numerical simulations to show that our hypothesis is justified in representative dynamical systems with three interacting frequencies: a quasiperiodically forced circle map, a pair of parametrically coupled forced nonlinear oscillators, and an experimental system consisting of an electronic circuit of forced phaselocked loops. Our observations lead us to conjecture that the ordering we predict may be universal in a large class of dynamical systems with three interacting frequencies. Firstly, we revise continued fractions and the Farey tree for the case of two frequencies. Consider a two-frequency system with autonomous frequency f 0 and external frequency f 1 . Let f ˜ 5 f 1 / f 0 . The aim is to define a sequence of

Error Detection and Correction Codes

In this article, we show how a new mathematical model of the genetic code can be exploited for investigating the almost periodic properties of DNA and mRNA protein-coding sequences. We present the main mathematical features of the model and highlight its connections with both number theory and group theory. The group theoretic framework presents interesting analogies with the theory of crystals. Moreover, we exploit the information provided by dichotomic classes, binary variables naturally derived from the mathematical model, in order to build statistical classifiers for retrieving and predicting the normal reading frame used by the ribosome in protein synthesis. The results show that coding sequences possess a local informational structure that can be related to frame synchronization processes. The information for retrieving the normal reading frame, which implies the existence of short-range correlations and almost periodic structures related to the organization of codons, offers an interesting analogy with the properties of quasi-crystals. From a theoretical point of view, our results might contribute to clarifying the relation between biological information and shape in nucleic acids and proteins. Also, from the point of view of applications, we present new promising tools for designing efficient algorithms for frame synchronization, which plays a crucial role in faithful synthesis of proteins.

The non-power model of the genetic code: a paradigm for interpreting genomic information.

In this article, we investigate the possible existence of error-detection/correction mechanisms in the genetic machinery by means of a recently proposed coding strategy. On this basis, we numerically code exons, creating binary parity strings and successively we study their dependence structure by means of rigorous statistical methods (moving block bootstrap, and a new entropy-based method). The results show that parity sequences display complex dependence patterns enforcing the hypothesis of the existence of deterministic error-correction mechanisms grounded on this particular parity coding.