Maximal Codes Research Articles

Site-specific recombinatorics: in situ cellular barcoding with the Cre Lox system.

BackgroundCellular barcoding is a recently developed biotechnology tool that enables the familial identification of progeny of individual cells in vivo. In immunology, it has been used to track the burst-sizes of multiple distinct responding T cells over several adaptive immune responses. In the study of hematopoiesis, it revealed fate heterogeneity amongst phenotypically identical multipotent cells. Most existing approaches rely on ex vivo viral transduction of cells with barcodes followed by adoptive transfer into an animal, which works well for some systems, but precludes barcoding cells in their native environment such as those inside solid tissues.ResultsWith a view to overcoming this limitation, we propose a new design for a genetic barcoding construct based on the Cre Lox system that induces randomly created stable barcodes in cells in situ by exploiting inherent sequence distance constraints during site-specific recombination. We identify the cassette whose provably maximal code diversity is several orders of magnitude higher than what is attainable with previously considered Cre Lox barcoding approaches, exceeding the number of lymphocytes or hematopoietic progenitor cells in mice.ConclusionsIts high diversity and in situ applicability, make the proposed Cre Lox based tagging system suitable for whole tissue or even whole animal barcoding. Moreover, it can be built using established technology.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0290-3) contains supplementary material, which is available to authorized users.

Codon Distribution in Error-Detecting Circular Codes.

In 1957, Francis Crick et al. suggested an ingenious explanation for the process of frame maintenance. The idea was based on the notion of comma-free codes. Although Crick’s hypothesis proved to be wrong, in 1996, Arquès and Michel discovered the existence of a weaker version of such codes in eukaryote and prokaryote genomes, namely the so-called circular codes. Since then, circular code theory has invariably evoked great interest and made significant progress. In this article, the codon distributions in maximal comma-free, maximal self-complementary and maximal self-complementary circular codes are discussed, i.e., we investigate in how many of such codes a given codon participates. As the main (and surprising) result, it is shown that the codons can be separated into very few classes (three, or five, or six) with respect to their frequency. Moreover, the distribution classes can be hierarchically ordered as refinements from maximal comma-free codes via maximal self-complementary codes to maximal self-complementary circular codes.

On the decomposition of prefix codes

In this paper we focus on the decomposition of rational and maximal prefix codes. We present an effective procedure that allows us to decide whether such a code is decomposable. In this case, the procedure also produces the factors of some of its decompositions. We also give partial results on the problem of deciding whether a rational maximal prefix code decomposes over a finite prefix code.

Maximal dinucleotide comma-free codes

The problem of retrieval and maintenance of the correct reading frame plays a significant role in RNA transcription. Circular codes, and especially comma-free codes, can help to understand the underlying mechanisms of error-detection in this process. In recent years much attention has been paid to the investigation of trinucleotide circular codes (see, for instance, Fimmel et al., 2014; Fimmel and Strüngmann, 2015a; Michel and Pirillo, 2012; Michel et al., 2012, 2008), while dinucleotide codes had been touched on only marginally, even though dinucleotides are associated to important biological functions. Recently, all maximal dinucleotide circular codes were classified (Fimmel et al., 2015; Michel and Pirillo, 2013). The present paper studies maximal dinucleotide comma-free codes and their close connection to maximal dinucleotide circular codes. We give a construction principle for such codes and provide a graphical representation that allows them to be visualized geometrically. Moreover, we compare the results for dinucleotide codes with the corresponding situation for trinucleotide maximal self-complementary C3-codes. Finally, the results obtained are discussed with respect to Crick׳s hypothesis about frame-shift-detecting codes without commas.

Maximal dinucleotide and trinucleotide circular codes

We determine here the number and the list of maximal dinucleotide and trinucleotide circular codes. We prove that there is no maximal dinucleotide circular code having strictly less than 6 elements (maximum size of dinucleotide circular codes). On the other hand, a computer calculus shows that there are maximal trinucleotide circular codes with less than 20 elements (maximum size of trinucleotide circular codes). More precisely, there are maximal trinucleotide circular codes with 14, 15, 16, 17, 18 and 19 elements and no maximal trinucleotide circular code having less than 14 elements. We give the same information for the maximal self-complementary dinucleotide and trinucleotide circular codes. The amino acid distribution of maximal trinucleotide circular codes is also determined.

Circular code motifs near the ribosome decoding center

A maximal C3 self-complementary trinucleotide circular code X is identified in genes of bacteria, eukaryotes, plasmids and viruses (Michel, 2015; Arquès and Michel, 1996). A translation (framing) code based on the circular code was proposed in Michel (2012) with the identification of several X circular code motifs (X motifs shortly) in both ribosomal RNAs (rRNAs) and their decoding center, and transfer RNAs (tRNAs). We extended these results in two ways. First, three universal X motifs were determined in the ribosome decoding center: the X motif mAA containing the conserved nucleotides A1492 and A1493, the X motif mG containing the conserved nucleotide G530 and the X motif m with unknown biological function (El Soufi and Michel, 2014). Secondly, statistical analysis of X motifs of greatest lengths performed on different and large tRNA populations according to taxonomy, tRNA length and tRNA score showed that these X motifs have occurrence probabilities in the 5′ and/or 3′ regions of 16 isoaccepting tRNAs of prokaryotes and eukaryotes greater than the random case (Michel, 2013). We continue here the previous works with the identification of X motifs in rRNAs of prokaryotes and eukaryotes near the ribosome decoding center. Seven X motifs PrRNAXm conserved in 16S rRNAs of prokaryotes P and four X motifs ErRNAXm conserved in 18S rRNAs of eukaryotes E are identified near the ribosome decoding center. Furthermore, four very large X motifs of length greater than or equal to 20 nucleotides, 14 large X motifs of length between 16 and 19 nucleotides and several X motifs of length greater or equal to 9 nucleotides are found in tRNAs of prokaryotes. Some properties of these X motifs in tRNAs are described. These new results strengthen the concept of a translation code based on the circular code (Michel, 2012).

Maximal noiseless code rates for collective rotation channels on qudits

We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set ${\mathcal E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm{SU}}(d)\}.$ This is done by analyzing the decomposition of the algebra ${\mathcal A}(d,n)$ generated by ${\mathcal E}(d,n)$. We summarize the results for the channels on qubits ($d=2$), and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general $d$. In particular, it is shown that the code rate, i.e., the number of protected qudits over the number of physical qudits, always approaches 1 for a suitable noiseless subsystem. Moreover, one can determine the maximum dimension of the noiseless subsystem by solving a non-trivial discrete optimization problem. The maximum dimension of the noiseless subsystem for $d = 3$ (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica.

Optimum Power Control at Finite Blocklength

This paper investigates the maximal channel coding rate achievable at a given blocklength $n$ and error probability $\epsilon$, when the codewords are subject to a long-term (i.e., averaged-over-all-codeword) power constraint. The second-order term in the large-$n$ expansion of the maximal channel coding rate is characterized both for additive white Gaussian noise (AWGN) channels and for quasi-static fading channels with perfect channel state information available at both the transmitter and the receiver. It is shown that in both cases the second-order term is proportional to $\sqrt{n^{-1}\ln n}$. For the quasi-static fading case, this second-order term is achieved by truncated channel inversion, namely, by concatenating a dispersion-optimal code for an AWGN channel subject to a short-term power constraint, with a power controller that inverts the channel whenever the fading gain is above a certain threshold. Easy-to-evaluate approximations of the maximal channel coding rate are developed for both the AWGN and the quasi-static fading case.

On the Bounds of Certain Maximal Linear Codes in a Projective Space

The set of all subspaces of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> is denoted by P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n). The subspace distance d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sub> (X, Y) = dim(X) + dim(Y) - 2 dim(X ∩ Y) defined on P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n) turns it into a natural coding space for error correction in random network coding. A subset of P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n) is called a code and the subspaces that belong to the code are called codewords. Motivated by classical coding theory, a linear coding structure can be imposed on a subset of P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n). Braun et al. conjectured that the largest cardinality of a linear code, that contains F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , is 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . In this paper, we prove this conjecture and characterize the maximal linear codes that contain F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> .

Structure and properties of strong prefix codes of pictures

A setX⊆ Σ** of pictures is a code if every picture over Σ is tilable in at most one way with pictures inX. The definition ofstrong prefix codeis introduced. The family of finite strong prefix codes is decidable and it has a polynomial time decoding algorithm. Maximality for finite strong prefix codes is also studied and related to the notion of completeness. We prove that any finite strong prefix code can be embedded in a unique maximal strong prefix code that has minimal size and cardinality. A complete characterization of the structure of maximal finite strong prefix codes completes the paper.

The maximal C3 self-complementary trinucleotide circular code X in genes of bacteria, eukaryotes, plasmids and viruses

In 1996, a set X of 20 trinucleotides is identified in genes of both prokaryotes and eukaryotes which has in average the highest occurrence in reading frame compared to the two shifted frames (Arquès and Michel, 1996). Furthermore, this set X has an interesting mathematical property as X is a maximal C3 self-complementary trinucleotide circular code (Arquès and Michel, 1996). In 2014, the number of trinucleotides in prokaryotic genes has been multiplied by a factor of 527. Furthermore, two new gene kingdoms of plasmids and viruses contain enough trinucleotide data to be analysed. The approach used in 1996 for identifying a preferential frame for a trinucleotide is quantified here with a new definition analysing the occurrence probability of a complementary/permutation (CP) trinucleotide set in a gene kingdom. Furthermore, in order to increase the statistical significance of results compared to those of 1996, the circular code X is studied on several gene taxonomic groups in a kingdom. Based on this new statistical approach, the circular code X is strengthened in genes of prokaryotes and eukaryotes, and now also identified in genes of plasmids. A subset of X with 18 or 16 trinucleotides is identified in genes of viruses. Furthermore, a simple probabilistic model based on the independent occurrence of trinucleotides in reading frame of genes explains the circular code frequencies and asymmetries observed in the shifted frames in all studied gene kingdoms. Finally, the developed approach allows to identify variant X codes in genes, i.e. trinucleotide codes which differ from X. In genes of bacteria, eukaryotes and plasmids, 14 among the 47 studied gene taxonomic groups (about 30%) have variant X codes. Seven variant X codes are identified with at least 16 trinucleotides of X. Two variant X codes XA in cyanobacteria and plasmids of cyanobacteria, and XD in birds are self-complementary, without permuted trinucleotides but non-circular. Five variant X codes XB in deinococcus, plasmids of chloroflexi and deinococcus, mammals and kinetoplasts, XC in elusimicrobia and apicomplexans, XE in fishes, XF in insects, and XG in basidiomycetes and plasmids of spirochaetes are C3 self-complementary circular. In genes of viruses, no variant X code is found.

Design of a Multiorder OFDM Frequency Diversity Approach

Frequency diversity is used to reduce the effect of destructive fading, so as to improve the communication quality, by passing the information symbols through multiple independently faded paths, and ensure that reliable communication is possible as long as one of these paths is strong. In this paper a multiorder orthogonal frequency division multiplexing (OFDM) frequency diversity approach using properties of order theory and Hamming distance is proposed. The frequency diversity is obtained by specifying proper correlations among the transmitted symbols. As subchannels experience independent fading, at least one of the symbols may have robust signal, which can be used by the receiver to detect other symbols. Considering bit error rate (BER) performance, power consumption, bandwidth utilization, and practical implementation expense, simulation results show that the proposed approach outperforms other OFDM diversity techniques, such as Maximal Ratio Combination (MRC) and Space Frequency Block Coding (SFBC).

On the hierarchy of trinucleotide n-circular codes and their corresponding amino acids

Circular codes are putative remnants of primeval comma-free codes and are potentially involved in detecting and maintaining the normal reading frame in protein coding sequences. In Michel and Pirillo (2013a) it was shown by computer algorithm that no maximal trinucleotide circular code can encode more than 18 different amino acids under the standard version of the genetic code. For comma-free codes the maximum is even less, namely 13 (Michel, 2014). The main purpose of this paper is to investigate these facts from a mathematical point of view and to show why the codes with the best-known error detecting properties are limited in the number of amino acids they can encode. We introduce five hierarchically ordered classes of trinucleotide codes including the well-known comma-free and circular codes and prove combinatorically that it is impossible to encode all amino acids using codes from four out of the five classes that have the strongest error detecting properties. However, it is possible to encode all 20 amino acids using codes from the largest class with the weakest properties. Additionally, we develop a handy criterion for circularity. As an application, it is shown that all codes from a special class of trinucleotide codes which includes the RNY-primeval code (Shepherd, 1986) are automatically circular. We also list which amino acids these codes encode.

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 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.

A note on the factorization conjecture

We give partial results on the factorization conjecture on codes proposed by Schutzenberger. We consider finite maximal codes C over the alphabet A = {a, b} with C \cap a^* = a^p, for a prime number p. Let P, S in Z <A>, with S = S_0 + S_1, supp(S_0) \subset a^* and supp(S_1) \subset a^*b supp(S_0). We prove that if (P,S) is a factorization for C then (P,S) is positive, that is P,S have coefficients 0,1, and we characterize the structure of these codes. As a consequence, we prove that if C is a finite maximal code such that each word in C has at most 4 occurrences of b's and a^p is in C, then each factorization for C is a positive factorization. We also discuss the structure of these codes. The obtained results show once again relations between (positive) factorizations and factorizations of cyclic groups.

New optimal constructions of conflict-avoiding codes of odd length and weight 3

Conflict-avoiding codes (CACs) have played an important role in multiple-access collision channel without feedback. The size of a CAC is the number of codewords which equals the number of potential users that can be supported in the system. A CAC with maximal code size is said to be optimal. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the maximal sizes of both equidifference and non equidifference CACs of odd prime length and weight 3 are obtained. Meanwhile, the optimal constructions of both equidifference and non equidifference CACs are presented. The numbers of equidifference and non equidifference codewords in an optimal code are also obtained. Furthermore, a new modified recursive construction of CACs for any odd length is shown. Non equidifference codes can be constructed in this method.

Permutation Cellular Automata

Permutation cellular automata are cellular automata defined by using finite maximal prefix codes. The overall dynamics of onesided and twosided permutation cellular automata is studied. For some classes of permutation cellular automata including the class of those defined by using finite maximal bifix codes, the overall dynamics is completely or partially described in terms of the codes used to define them.

Maximally-dense-coding-capable quantum states

A complementarity relation is established between the capacity of multiport classical information transmission via quantum states and multiparty quantum correlation measures for three-qubit pure states. The multiparty quantum correlation measures considered are the generalized geometric measure, the tangle, and the discord monogamy score. The complementarity relation is revealed by the identification of a one-parameter family of pure three-qubit states, which we call the maximally dense coding capable family of states. These states have the maximal multiport dense coding capacity among all three-qubit pure states with an arbitrary fixed amount of the multiparty quantum correlations.

FULL MONOIDS AND MAXIMAL CODES

In recent years codes that are not Uniquely Decipherable (UD) were studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a free monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal set. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code. Finally we generalize some properties of UD codes.

SOME PROPERTIES OF TRINUCLEOTIDES AND TETRANUCLEOTIDES

In 1994, with a statistical study of trinucleotide occurences per frame, D. Arquès and C. Michel identified the fol lowing set of 20 trin- ucleotides in the gene population of both eucaryotes E U K and procaryotes P RO: {AAC , AAT , AC C , ATC,ATT,CAG,CTC,CTG,GAA,GAC,GAG,GAT, GCC,GGC,GGT,GTA,GTC,GTT,TAC,TTC}.Thisset of words of length three on the genetic alphabet A4 = {A, C, G, T } has remarkable properties: it is a self-complementary set, it is a circular code, it is maximal and it has the C3- property. This and other identifications of trinucleotide circular codes in different genomes in the last twenty years raised interest in the concept of trinucleotide circular code for genetics. In 2003 we found an efficient algorithm for testing the circularity of a trinucleotide code and in 2005 we found the list of al l 528 maximal self-complementary circular codes. In 2008 we presented a hierarchy of the self-complementary circular codes and in 2012 we presented a hierarchy of al l circular codes. These circular codes could permit the identification, either in paral lel with or substituting existing methods used by biologist, of as yet unknown coding regions of DNA. A hierarchy of tetranucleotide circular codes is one of our aims in the future. In this paper we begin the study of the unbordered tetranucleotides and of the &ldquo;forbidden configurations&rdquo; for tetranucleotides and we give a first result.