| Summary: | The investigation of DNA has been one of the most developed areas of research in this and in the last century. However, there is a long way to go to fully understand the DNA code. With the increasing of DNA sequenced data, mathematical methods play an important role in addressing the need for e cient quantitative techniques for the detection of regions of interest and overall characteristics in these sequences. A feature of interest in the study of genomic words is their spatial distribution along a DNA sequence, which can be characterized by the distances between words. Counting such distances provides discrete distributions that may be analyzed from a statistical point of view. In this work we explore the distances between genomic words as a mathematical descriptor of DNA sequences. The main goal is to design, develop and apply statistical methods specially designed for their distributions, in order to capture information about the primary and secondary structure of DNA. The characterization of empirical inter-word distance distributions involves the problem of the exponential increasing of the number of distributions as the word length increases, leading to the need of data reduction. Moreover, if the data can be validly clustered, the class labels may provide a meaningful description of similarities and di erences between sets of distributions. Therefore, we explore the inter-word distance distributions potential to obtain a word clustering, able to highlight similar patterns of word distributions as well as summarized characteristics of each set of distributions. With the aim of performing comparative studies between genomic sequences and de ning species signatures, we deduce exact distributions of inter-word distances under random scenarios. Based on these theoretical distributions, we de ne genomic signatures of species able to discriminate between species and to capture their evolutionary relation. We presume that the study of distributions similarities and the clustering procedure allow identifying words whose distance distribution strongly di ers from a reference distribution or from the global behaviour of the majority of the words. One of the key topics of our research focuses on the establishment of procedures that capture distance distributions with atypical behaviours, herein referred to as atypical distributions. In the genomic context, words with an atypical distance distribution may be related with some biological function (motifs). We expect that our results may be used to provide some sort of classi cation of sequences, identifying evolutionary patterns and allowing for the prediction of functional properties, thereby contributing to the advancement of knowledge about DNA sequences.
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