Finally, the set of symmetry descriptors are tested on binarized 1 / f noises and binarized DNA sequences in Section 4 then, the results are discussed in Section 5.įor a binary sequence, an m-palindrome is, by definition, a grouping of m bits that form an m-pattern of mirror symmetry. Then, the concept of palindromes is extended and new tools such as symmetropy and symmentropy are proposed in Section 3. To reach this goal, the concept of palindrome and then that of frieze is presented in Section 2. If the objective is indeed to extend the analysis of discrete periodic sequences to other types of sequences, then the search for all symmetric patterns is the next step. As an illustration, five types of alphabetical sequences of 16 characters, having the same symmetries as friezes, are presented as follows: b b b b b b b b b b b b b b b b, d b d b d b d b d b d b d b d b, b p b p b p b p b p b p b p b p, b q b q b q b q b q b q b q b q, b q p d b q p d b q p d b q p d. Wanting to extract many more intrinsic features in the discrete sequences studied can consist of looking for other types of symmetries, as it is explicitly the case in friezes.Ī frieze is a horizontal strip composed of an infinite number of symmetrical patterns, i.e., a periodic geometric object. Today, when studying a word or a discrete sequence, its analysis is still limited to only one type of symmetry: the “mirror” symmetry. , which is used as a starting point in this work and in particular the notion of palindromic complexity. Even if the theoretical research around the palindrome is still going on, as shown by the recent article by Gabric and Shallit to name but a few, it is the older work of Allouche et al. For example, the alphabetic character sequence d d d d d d d d b b b b b b b is a 16-palindrome composed of two 8-patterns: d d d d d d d d, b b b b b b b b. To fix ideas, a palindrome of size m, called “ m-palindrome”, is a discrete sequence composed of two contiguous symmetrical (mirror) sub-sequences each composed of k-patterns with k = ⌊ m / 2 ⌋. Note that ‘ T’ is the complementary of ‘ A’, and ‘ C’ is the complementary of ‘ G’). It is composed of a 4-pattern on the right (‘ C C A T’) obtained by a mirror reflection of its 4-pattern complementary on the left (‘ A T G G’): ‘ A T G G’|‘ C C A T’, where It is qualified as a 8-palindrome sequence. (Let us consider the sequence of characters ‘ A T G G C C A T’. Its greatest success is undoubtedly derived from the analysis of biological sequences (DNA, RNA and proteins), even if in this case the definition of DNA palindromes is slightly different from the classical definition. The “mirror” symmetry on which the concept of palindrome was based is certainly the basis of the oldest symmetry descriptors. In the continuity of the work carried out by Tibatan and Sarisaman, our article aims to highlight the symmetry links between the concept of frieze and the concept of palindrome, which have been insufficiently exploited until now in the analysis of binary data. Very recently, the study of quantum behavior, encountered in palindromes within the DNA structure, revealed that the symmetry properties of the unitary structure, other than those present in classical palindromes, play an important role in the origin and cause of mutations. The palindromic analysis of discrete sequences has partly revolutionized molecular biology and is widely used as shown by the following work, to name a few. Moreover, it is highlighted that a certain number of these new palindromes of sizes greater than 30 bits are more discriminating than those of smaller sizes assimilated to those from an independent and identically distributed random variable. A factor of 4 between the slopes obtained from the linear fits of the local symmentropies for the two DNA sequences shows the discriminative capacity of the local symmentropy. A relative error of 6 % of symmetropy is obtained from the HUMHBB and YEAST1 DNA sequences. These new palindromes with new types of symmetry also allow for better discrimination of binarized DNA sequences. New tools, such as symmetropy and symmentropy, based on new types of palindromes allow us to discriminate binarized 1 / f noise sequences better than Lempel–Ziv complexity. However, other types of symmetry, such as those present in friezes, could allow us to analyze binary sequences from another point of view. Today, the palindromic analysis of biological sequences, based exclusively on the study of “mirror” symmetry properties, is almost unavoidable.
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