# Global Repeat Map (GRM) Application: Finding All DNA Tandem Repeat Units
## 1 Introduction  
Repetitive DNA is an important component of eukaryotic genomes, comprising more than 50% of the genome size in most species [1, 2]. It is composed of a large number of different classes of repetitive DNA sequences, either dispersed or arranged in tandem [3, 4, 2]. Among tandem repetitive DNA, there are moderately repetitive DNAs, such as ribosomal RNA (rRNA) and protein-coding gene families or short tandem telomeric repeats, as well as highly repetitive non-coding microsatellite and satellite sequences [3]. Tandem repeats (TRs) have a propensity to mutate and become polymorphic by expansion or contraction in the number of repeat units. This may be due to slippage during DNA replication, unequal crossing-over during recombination, or the imprecise repair of double-strand DNA breaks [5, 6, 7]. Tandem repeats appear in two different structural forms: as the TRs of individual head-to-tail monomers, forming monomeric arrays, and as the TRs of numer higher order repeats (HOR): copies consisting of TRs of \(n\) head-to-tail monomers. Divergence among HOR copies in a HOR array is small, below \(\sim\)5%, while divergence among monomers within each HOR copy is sizably larger, 20-40% [8; 9; 10]. The significance of repetitive DNA in the genome is not completely understood, and it has been considered to have both structural and functional roles. Various TR arrays are generally found in heterochromatin and form essential chromosome structures such as centromeres and telomeres [11; 12]. Tandem repeat DNAs are implicated in centromeric functions, such as segregation in mitosis and meiosis, essential during cell division, the pairing of homologous chromosomes, sister chromatid attachment and the formation of kinetochore structures. In addition, the most interesting potential role of TRs is gene regulation, related to a general concept that the regulatory system of genomes is encoded in networks of repetitive sequence relationships [13; 14]. If present in upstream regulatory regions, TRs can modify the chromatin structure and affect transcription, resulting in altered gene expression and protein abundance [5; 15; 16; 17; 18; 19; 20]. In this sense, there are established links between TRs and phenotypes in a wide range of organisms, including humans [21]. For example, TR derepression is associated with cancer outcomes [22], chromosome mis-segregation and aneuploidy [23] and aging [24]. In addition, a variation in TR copy number has been associated with genetic incompatibilities between species [25], evolutionary development [26; 27; 28] and with different diseases, such as fragile-X mental retardation [29], [PERSON]'s disease [30], myotonic dystrophy [31], spinal and bulbar muscular atrophy [32] and [PERSON]'s ataxia [33]. Given the importance of the known and potential biological roles of TRs and their usefulness in other biological studies, the identification and analysis of all types of TRs are currently of substantial interest. From this perspective, it seemed essential to us to develop an efficient and sensitive algorithm for the identification of all kinds of TRs in various genomic sequences. A vast range of algorithms already exists for the direct or indirect detection of TRs in DNA sequence [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. In general, all TR detection algorithms can be divided into two main categories: flexible statistical string-matching algorithms, and signal processing algorithms. Although these algorithms use a whole range of different methods and powerful analysis tools developed in traditional signal processing, all suffer from significant limitations. Some of the relevant observed problems that are analyzed and pointed out in Refs. [36; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51] could be summarized as follows.