Absent words and the (dis)similarity analysis of DNA sequences: an experimental study
 Mohammad Saifur Rahman^{1},
 Ali Alatabbi^{2},
 Tanver Athar^{2},
 Maxime Crochemore^{2, 3} and
 M. Sohel Rahman^{1}Email authorView ORCID ID profile
Received: 22 July 2015
Accepted: 3 March 2016
Published: 22 March 2016
Abstract
Background
An absent word with respect to a sequence is a word that does not occur in the sequence as a factor; an absent word is minimal if all its factors on the other hand occur in that sequence. In this paper we explore the idea of using minimal absent words (MAW) to compute the distance between two biological sequences. The motivation and rationale of our work comes from the potential advantage of being able to extract as little information as possible from large genomic sequences to reach the goal of comparing sequences in an alignmentfree manner.
Findings
We report an experimental study on the use of absent words as a distance measure among biological sequences. We provide recommendations to use the best index based on our analysis. In particular, our analysis reveals that the best performers are: the length weighted index of relative absent word sets, the length weighted index of the symmetric difference of the MAW sets, and the Jaccard distance between the MAW sets. We also found that during the computation of the absent words, the reverse complements of the sequences should also be considered.
Conclusion
The use of MAW to compute the distance between two biological sequences has potential advantage over alignment based methods. It is expected that this potential advantage would encourage researchers and practitioners to use this as a (dis)similarity measure in the context of sequence comparison and phylogeny reconstruction. Therefore, we present here a comparison among different possible models and indexes and pave the path for the biologists and researchers to choose an appropriate model for such comparisons.
Keywords
Findings
Background
Recently, the concept of minimal absent word (MAW) has been used to compute the distance between two species [1]. Similar effort has also been made to investigate the variation in number and content of MAWs within a species using four human genome assemblies [2]. This concept along with the related notions of absent words, also known as nullomers and forbidden words, have received significant attention in the relevant literature (e.g., [3–11]) and have been shown to be useful in applications like text compression [12,13]. Perhaps the most significant use of this concept is in the field of computational biology. Hampikian and Andersen have studied nullomers, i.e., the shortest words that do not occur in a given genome, and primes, i.e., the shortest words that are absent from the entire known genetic data with a motivation to discover the constraints on natural DNA and protein sequences [14]. Acquisti et al. [15] have studied nullomers and the cause of absent words in the human genome. Herold et al. [16] have presented a method to compute the shortest absent words in genomic sequences. Pinho et al. [17] on the other hand focused on MAWs that form a set smaller than the set of absent words. Subsequently, Garcia and Pinho have studied four human genome assemblies from the perspective of MAWs [2].
The main focus of this paper is to study and analyze possible indexes that can be used with MAWs to establish an alignmentfree distance or similarity measure. The motivation and rationale of using MAW comes from the potential advantage of being able to extract as little information as possible from large genomic sequences to reach the goal of comparing them with one another. And this has recently attracted researchers to propose distance measures based on MAWs. For example, in [1], Chairungsee and Crochemore have proposed a distance measure based on the set of MAWs and have used that distance measure to construct a phylogenetic tree among 11 species, following an experimental setup of Liu and Wang [18]. And, in [2], Garcia and Pinho have explored the potential of the MAWs from the perspective of similarities and differences among 4 human genome assemblies.
Indexes used and compared in this paper as a distance/similarity measure
Index  Comment 

Lengthweighted index (LWI)  Considered in [1] for only symmetric difference. Here we also use it for set intersection 
Jaccard distance  Used in this paper 
Total variation distance (TVD)  Used in [2] to analyze similarity on four human genome assemblies 
GC content  Used in [2] to analyze similarity on four human genome assemblies. Here we use GC content on symmetric difference, set intersection of MAW sets as well as on RAW sets 
Relative absent word (RAW)  Considered in [20] to study Ebola virus genomes against human DNA. Here we use RAW sets for LWI and GC content measures 
Methods
A string \(x = x_1, x_2, \ldots , x_n\) is a sequence of characters of length n from a finite alphabet \(\Sigma\), i.e., \(x_i \in \Sigma , 1\le i\le n\). An empty string is denoted by \(\epsilon\). A string y is a factor or substring of a string x iff there exist strings u, v such that \(x=uyv\); if \(u\ne \epsilon\) or \(v\ne \epsilon\), then, y is a proper factor of x. We use the term word and string synonymously. An absent word in a string is a word that does not occur in the given string. More formally, a string y is an absent word in a string x if it is not a factor of x. Additionally, if all its proper factors are factors of x, then y is said to be a MAW. For example, aaa, aba, and bbb are examples of MAWs for the string \(x = abbaab\). But, aaab is an absent word but not a MAW of x. Given a string x, we will use \(MAW_x\) to denote the set of MAWs of x.
 Step 1:
For each sequence \(s_i, 1\le i\le k\), we compute \(MAW_{s_i}\).
 Step 2:
We compute distance matrix \(\mathcal M_{\mathcal S}^{\mathcal D}\) for the set \(\mathcal S\) using a distance measure \(\mathcal D\) based on \(MAW_{s_i}, 1\le i\le k\). For all \(1\le i,j\le k\), we have \(\mathcal M_{\mathcal S}^{\mathcal D}[i,j] = \mathcal D[s_i, s_j]\). Because the distance measure is symmetric, we need only focus on the upper triangle of the matrix \(\mathcal M_{\mathcal S}^{\mathcal D}\).
 Step 3:
We build a phylogenetic tree \(\mathcal T^{\mathcal D}_{\mathcal A}(\mathcal S)\) on the set \(\mathcal S\) based on the distance measure \(\mathcal D\) applying algorithm \(\mathcal A\) on \(\mathcal M_{\mathcal S}^{\mathcal D}\) for phylogeny reconstruction.
Distance measures
We apply a number of distance measures discussed below. In what follows we will consider two sequences x and y and their MAW sets, \(MAW_{x}\) and \(MAW_{y}\).
Lengthweighted index
Jaccard distance
Total variation distance (TVD)
GC content
Relative absent words (RAWs)
Results and discussion
We have used the same datasets used in [18] and [1]. In particular, we have conducted our experiments on the first exon sequences of \(\beta\)globin genes from 11 species, namely, Human, Goat, Gallus, Opossum, Lemur, Mouse, Rabbit, Rat, Bovine, Gorilla, and Chimpanzee. Because the gene family of \(\beta\)globin has a significant biological role in oxygen transport in organisms, it is used to analyze DNA and the first exon of the \(\beta\)globin gene is an example for many DNA studies instead of computing similarity/dissimilarity of the whole genomes. Inspired by the experimental setup of Garcia and Pinho [2], we consider two scenarios: the original sequence itself and the original sequence concatenated with its reversed complement (artificial words across the boundary between both sequences are ignored). The former will be referred to as the noRC setting and the latter as the RC setting. The motivation for using the reverse complement is to take into consideration words that might occur in the reverse complement strand but that might be absent from the direct strand.
We have used the algorithm of [11] to compute the MAW sets using their implementation, which is available at: http://github.com/solonas13/maw. We have used EAGLE software of [20] to compute the RAW sets; EAGLE is available at: http://bioinformatics.ua.pt/software/eagle/. The code to compute the distance matrices and analyze the results were written in C++ language and can be found at: https://github.com/srautonu/AWorDS. We have also implemented a related webbased tool with limited capacity here: http://www.ekngine.com/AWorDS. It is planned that this webtool will be improved with more functionalities in near future.
We have considered five distance measures described in “Distance measures” section based on the MAW sets. Additionally, we have considered LWI and GC content distance measures involving RAW sets. With noRC and RC settings, this gives us a total of 14 distance matrices. For the sake of brevity we do no provide all the distance matrices in this paper. However, these can be found here: https://github.com/srautonu/AWorDS and also in the Additional files 1, 2, 3 and 4.
Discussion

It is believed that Gorilla and Chimpanzee are most similar to Human [REL 1];

Similarly, among these 11 species, Goat and Bovine should be similar [REL 2] as are Rat and Mouse [REL 3];

Gallus and Opossum should be remote from the other species because Gallus is the only nonmammalian representative in this group [REL 4] and Opossum is the most remote species from the remaining mammals [REL 5];

Besides gallus and Opossum, lemur is more remote from the other species relatively [REL 6].
We have analyzed the distance measures based on the abovementioned six expected relations (REL 1–REL 6). Among these six relations we give higher importance on REL 1 through REL 3 in the sense that when all of these are captured we look into the rest for further comparison. Below we discuss several interesting points from our analysis. Notably, we have provided a spreadsheet (Additional file 4) with a brief description of the content as a Additional files 1, 2, 3 and 4 that we have used for this analysis.

As is evident from our analysis, unfortunately, the GCC measure does not do very well in comparison to the other metrics despite that it is more related to the content of the minimal absent words. In particular, in most cases this measure is unable to capture the expected relationships (REL 1–REL 6) mentioned above. However, despite the overall relative poor performance, except for the cases when intersection operation has been used, GCC measure is at least able to capture the close relation among Human, Gorilla and Chimpanzee, i.e., REL 1. For intersection operation however, GCC fails miserably to capture any of the important relationships among REL 1 REL 2 and REL 3.

The TVD also fails to be highly impressive. It has been able to capture some of the relations but not all. However, it definitely seems better than the GCC measures. In particular, it has been able to capture REL 1 and in most cases it also captures REL 2. However, it fails to capture REL 3 in both RC and NoRC settings.
The distance matrix based on the length weighted index on RAW sets (on RC setting)
Species  Human  Goat  Opossum  Gallus  Lemur  Mouse  Rabbit  Rat  Gorilla  Bovine  Chimp 

Human  23.39  26.94  28.34  27.82  23.49  19.31  27.88  4.77  21.60  7.26  
Goat  28.71  24.16  25.89  25.52  24.33  27.43  21.77  8.73  24.26  
Opossum  29.55  31.23  29.21  26.69  30.52  26.90  28.16  28.44  
Gallus  28.66  30.22  26.27  30.89  28.25  26.21  30.51  
Lemur  30.21  27.63  30.96  27.77  25.91  30.27  
Mouse  24.09  26.43  20.98  23.17  23.29  
Rabbit  29.19  19.02  22.28  21.50  
Rat  28.37  27.95  30.21  
Gorilla  19.48  9.62  
Bovine  21.97  
Chimp 
The sorted list of each species from a particular species (left most column of each row) according to the computed distance based on the length weighted index on RAW sets (on RC setting)
Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Mouse  \(\rightarrow\)Opossum  \(\rightarrow\)Lemur  \(\rightarrow\)Rat  \(\rightarrow\)Gallus 
Goat  \(\rightarrow\)Bovine  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Gallus  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Mouse  \(\rightarrow\)Lemur  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Opossum  \(\rightarrow\)Rabbit  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Bovine  \(\rightarrow\)Chimp  \(\rightarrow\)Goat  \(\rightarrow\)Mouse  \(\rightarrow\)Gallus  \(\rightarrow\)Rat  \(\rightarrow\)Lemur 
Gallus  \(\rightarrow\)Goat  \(\rightarrow\)Bovine  \(\rightarrow\)Rabbit  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Lemur  \(\rightarrow\)Opossum  \(\rightarrow\)Mouse  \(\rightarrow\)Chimp  \(\rightarrow\)Rat 
Lemur  \(\rightarrow\)Goat  \(\rightarrow\)Bovine  \(\rightarrow\)Rabbit  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Gallus  \(\rightarrow\)Mouse  \(\rightarrow\)Chimp  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Mouse  \(\rightarrow\)Gorilla  \(\rightarrow\)Bovine  \(\rightarrow\)Chimp  \(\rightarrow\)Human  \(\rightarrow\)Rabbit  \(\rightarrow\)Goat  \(\rightarrow\)Rat  \(\rightarrow\)Opossum  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus 
Rabbit  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Bovine  \(\rightarrow\)Mouse  \(\rightarrow\)Goat  \(\rightarrow\)Gallus  \(\rightarrow\)Opossum  \(\rightarrow\)Lemur  \(\rightarrow\)Rat 
Rat  \(\rightarrow\)Mouse  \(\rightarrow\)Goat  \(\rightarrow\)Human  \(\rightarrow\)Bovine  \(\rightarrow\)Gorilla  \(\rightarrow\)Rabbit  \(\rightarrow\)Chimp  \(\rightarrow\)Opossum  \(\rightarrow\)Gallus  \(\rightarrow\)Lemur 
Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Bovine  \(\rightarrow\)Mouse  \(\rightarrow\)Goat  \(\rightarrow\)Opossum  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus  \(\rightarrow\)Rat 
Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Mouse  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Chimp  \(\rightarrow\)Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Rabbit  \(\rightarrow\)Bovine  \(\rightarrow\)Mouse  \(\rightarrow\)Goat  \(\rightarrow\)Opossum  \(\rightarrow\)Rat  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus 

Among the distance measures one of the best (if not the best) performers turns out to be the length weighted index applied on the RAW sets. In particular, Table 2 (also see Table 3) has all the desired relations (REL 1 through REL 6) mentioned above. As expected, the result is better when RC setting is used.
The distance matrix based on the Jaccard distance on MAW sets (on RC setting)
Species  Human  Goat  Opossum  Gallus  Lemur  Mouse  Rabbit  Rat  Gorilla  Bovine  Chimp 

Human  0.70  0.82  0.80  0.76  0.70  0.61  0.80  0.15  0.69  0.26  
Goat  0.84  0.74  0.74  0.77  0.77  0.79  0.69  0.36  0.71  
Opossum  0.85  0.87  0.91  0.84  0.90  0.82  0.85  0.82  
Gallus  0.81  0.82  0.79  0.85  0.80  0.81  0.80  
Lemur  0.83  0.81  0.81  0.76  0.72  0.77  
Mouse  0.78  0.78  0.64  0.74  0.68  
Rabbit  0.81  0.63  0.75  0.65  
Rat  0.80  0.82  0.82  
Gorilla  0.67  0.15  
Bovine  0.69  
Chimp 
The sorted list of each species from a particular species (left most column of each row) according to the computed distance based on the Jaccard distance on MAW sets (on RC setting)
Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Bovine  \(\rightarrow\)Mouse  \(\rightarrow\)Goat  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Goat  \(\rightarrow\)Bovine  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus  \(\rightarrow\)Rabbit  \(\rightarrow\)Mouse  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Opossum  \(\rightarrow\)Chimp  \(\rightarrow\)Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Rabbit  \(\rightarrow\)Goat  \(\rightarrow\)Gallus  \(\rightarrow\)Bovine  \(\rightarrow\)Lemur  \(\rightarrow\)Rat  \(\rightarrow\)Mouse 
Gallus  \(\rightarrow\)Goat  \(\rightarrow\)Rabbit  \(\rightarrow\)Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Chimp  \(\rightarrow\)Bovine  \(\rightarrow\)Lemur  \(\rightarrow\)Mouse  \(\rightarrow\)Opossum  \(\rightarrow\)Rat 
Lemur  \(\rightarrow\)Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Rat  \(\rightarrow\)Gallus  \(\rightarrow\)Mouse  \(\rightarrow\)Opossum 
Mouse  \(\rightarrow\)Gorilla  \(\rightarrow\)Chimp  \(\rightarrow\)Human  \(\rightarrow\)Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Rat  \(\rightarrow\)Rabbit  \(\rightarrow\)Gallus  \(\rightarrow\)Lemur  \(\rightarrow\)Opossum 
Rabbit  \(\rightarrow\)Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Chimp  \(\rightarrow\)Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Mouse  \(\rightarrow\)Gallus  \(\rightarrow\)Lemur  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Rat  \(\rightarrow\)Mouse  \(\rightarrow\)Goat  \(\rightarrow\)Human  \(\rightarrow\)Gorilla  \(\rightarrow\)Rabbit  \(\rightarrow\)Lemur  \(\rightarrow\)Chimp  \(\rightarrow\)Bovine  \(\rightarrow\)Gallus  \(\rightarrow\)Opossum 
Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Rabbit  \(\rightarrow\)Mouse  \(\rightarrow\)Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Chimp  \(\rightarrow\)Lemur  \(\rightarrow\)Mouse  \(\rightarrow\)Rabbit  \(\rightarrow\)Gallus  \(\rightarrow\)Rat  \(\rightarrow\)Opossum 
Chimp  \(\rightarrow\)Gorilla  \(\rightarrow\)Human  \(\rightarrow\)Rabbit  \(\rightarrow\)Mouse  \(\rightarrow\)Bovine  \(\rightarrow\)Goat  \(\rightarrow\)Lemur  \(\rightarrow\)Gallus  \(\rightarrow\)Opossum  \(\rightarrow\)Rat 

Jaccard distance has also turned out to be a very good measure in our experiments. In particular, in Table 4 (also see Table 5) we can identify almost all desired relations (REL 1 through REL 6).

Length Weighted Index (LWI) for symmetric difference under the RC setting also performs very well in conserving relations REL 1 through REL 5. This measure seems quite good under the NoRC setting as well. However, it is worthmentioning that under the latter setting it fails to capture the close relation between Rat and Mouse (REL 3).

In general it seems that the results are better for the RC setting which is expected because this setting takes into consideration words that might occur in the reverse complement strand but that might be absent from the direct strand.
Phylogenetic tree reconstruction
Recommendations

LWI applied on the RAW sets, the same, i.e., LWI applied on symmetric difference and Jaccard distance are the best performers and should be used in computing distance matrixes based on absent words.

RC setting should be preferable. This is supported by the natural assumption that this setting takes into consideration words that might occur in the reverse complement strand but that might be absent from the direct strand.
Availability of supporting data
The data used in our experiments, the code to compute the distance matrices and analyze the results can be found here: https://github.com/srautonu/AWorDS. The implementation of the algorithm of [11] is available here: http://github.com/solonas13/maw. The EAGLE software of [20] to compute the RAW sets is available here: http://bioinformatics.ua.pt/software/eagle/. We have also setup a preliminary version of a webbased tool here: http://www.ekngine.com/AWorDS.
Declarations
Authors’ contributions
MSR (Sohel) and MC conceived of the study. MSR (Saifur), MC and MSR (Sohel) Designed the experiments. MSR (Saifur) wrote the codes. MSR (Saifur) and TA ran the experiments. MSR (Saifur) and MSR (Sohel) Analyzed the results. AA and MSR (Saifur) implemented the online tool. MSR (Sohel) supervised and coordinated the work and wrote the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This research work has been partially supported by an INSPIRE Strategic Partnership Award, administered by the British Council, Bangladesh for the project titled “Advances in algorithms for next generation biological sequences”. Tanver Athar is supported by an EPSRC grant (Doctoral Training Grant #EP/L504798/1). The authors acknowledge the critiques of the anonymous reviewers and the comments from Editors that helped improve the manuscript. Part of this research work was done when M. Sohel Rahman was on a Sabbatical Leave from BUET and was visiting King’s College, London.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
 Chairungsee S, Crochemore M. Using minimal absent words to build phylogeny. Theory Comput Sci. 2012;450:109–16. doi:10.1016/j.tcs.2012.04.031.View ArticleGoogle Scholar
 Garcia SP, Pinho AJ. Minimal absent words in four human genome assemblies. PLoS One. 2011;6(12):29344.View ArticleGoogle Scholar
 Béal M, Mignosi F, Restivo A. Minimal forbidden words and symbolic dynamics. In: STACS 96, 13th annual symposium on theoretical aspects of computer science. Grenoble: Proceedings. 1996. p. 555–66.Google Scholar
 Fici G, Mignosi F, Restivo A, Sciortino M. Word assembly through minimal forbidden words. Theory Comput Sci. 2006;359(1–3):214–30. doi:10.1016/j.tcs.2006.03.006.View ArticleGoogle Scholar
 Béal M, Fiorenzi F, Mignosi F. Minimal forbidden patterns of multidimensional shifts. IJAC. 2005;15(1):73–93. doi:10.1142/S0218196705002165.Google Scholar
 Mignosi F, Restivo A, Sciortino M. Words and forbidden factors. Theory Comput Sci. 2002;273(1–2):99–117. doi:10.1016/S03043975(00)004369.View ArticleGoogle Scholar
 Mignosi F, Restivo A, Sciortino M. Forbidden factors and fragment assembly. ITA. 2001;35(6):565–77. doi:10.1051/ita:2001132.Google Scholar
 Béal M, Crochemore M, Mignosi F, Restivo A, Sciortino M. Computing forbidden words of regular languages. Fundam Inf. 2003;56(1–2):121–35.Google Scholar
 Crochemore M, Mignosi F, Restivo A. Automata and forbidden words. Inf Process Lett. 1998;67(3):111–7. doi:10.1016/S00200190(98)001045.View ArticleGoogle Scholar
 Wu Z, Jiang T, Su W. Efficient computation of shortest absent words in a genomic sequence. Inf Process Lett. 2010;110(14–15):596–601. doi:10.1016/j.ipl.2010.05.008.View ArticleGoogle Scholar
 Barton C, Heliou A, Mouchard L, Pissis SP. Lineartime computation of minimal absent words using suffix array. BMC Bioinform. 2014;15:388. doi:10.1186/s1285901403889.View ArticleGoogle Scholar
 Crochemore M, Mignosi F, Restivo A, Salemi S. Text compression using antidictionaries. In: Automata, languages and programming, 26th international colloquium, ICALP’99, Prague: Proceedings. 1999. p. 261–70.Google Scholar
 Crochemore M, Navarro G. Improved antidictionary based compression. In: 22nd international conference of the Chilean computer science society (SCCC 2002). Copiapo; 2002. p. 7–13. doi:10.1109/SCCC.2002.1173168. http://doi.ieeecomputersociety.org/10.1109/SCCC.2002.1173168
 Hampikian G, Andersen TL. Absent sequences: nullomers and primes. In: Biocomputing 2007, Proceedings of the Pacific symposium. Maui: 2007. p. 355–66. http://psb.stanford.edu/psbonline/proceedings/psb07/hampikian
 Acquisti C, Poste G, Curtiss D, Kumar S. Nullomers: really a matter of natural selection? PLoS One. 2007;2(10):1022.View ArticleGoogle Scholar
 Herold J, Kurtz S, Giegerich R. Efficient computation of absent words in genomic sequences. BMC Bioinform. 2008;9:167. doi:10.1186/147121059167.View ArticleGoogle Scholar
 Pinho AJ, Ferreira PJSG, Garcia SP, Rodrigues JMOS. On finding minimal absent words. BMC Bioinform. 2009;10:137. doi:10.1186/1471210510137.View ArticleGoogle Scholar
 Liu N, Wang TM. A relative similarity measure for the similarity analysis of DNA sequences. Chem Phys Lett. 2005;408(4):307–11.View ArticleGoogle Scholar
 Dembo A, Karlin S. Poisson approximations for rscan processes. Ann Appl Probab. 1992;2(2):329–57.View ArticleGoogle Scholar
 Silva RM, Pratas D, Castro L, Pinho AJ, Ferreira PJ. Three minimal sequences found in ebola virus genomes and absent from human DNA. Bioinformatics. 2015;31:2421.View ArticlePubMedPubMed CentralGoogle Scholar
 Sung WK. Algorithms in Bioinformatics: A Practical Introduction. USA: CRC Press; 2011.Google Scholar
 Saitou N, Nei M. The neighborjoining method: A new method for reconstructing phylogenetic trees. Journal of Molecular Biology and Evolution. 1987;4(4):406–25.PubMedGoogle Scholar