The reconstruction of a phylogenetic tree is a primary problem in evolutionary biology. Sequence alignment is a key step in the reconstruction, aiming to identify the homology of sequences and uncover phylogenetic relationships in sequences. Traditional sequence comparison is based on pairwise or multiple sequence alignment (Felsenstein and Felenstein, 2004; Morrison, 2006) and was implemented by software packages such as BLAST (Altschul et al., 1990), ClustalW (Thompson et al., 1994), and MrBayes (Ronquist et al., 2012). However, the methods based on sequence alignment have some disadvantages, including high computational cost in handling the time and space complexity of the algorithm. Therefore, alignment-free methods have been proposed to overcome these problems (Zielezinski et al., 2017). The computational cost of alignment-free methods is low because they are generally of linear complexity (Fox et al., 1977).

Several alignment-free methods for sequence comparison are based on word counts (Blaisdell, 1986; Höhl et al., 2006; Wang et al., 2016). A key idea is to use the close distribution of k-mers to imply the high correlation degree, hence the similarity of the sequences. The methods have been implemented in software tools, such as FFP (Sims et al., 2009), kWIP (Murray et al., 2017), CVtree (Qi et al., 2004), and DLtree (Wu et al., 2017). Many k-mer methods transform the input sequence into a frequency vector of k-mers, then define the distance of the sequences by that of the frequency vector of k-mers (Qi et al., 2004; Wu et al., 2017). To reduce the statistical dependence between adjacent word matches, Spaced-Words (Leimeister and Boden, 2014) proposed to use spaced words, which are defined by patterns of matches without reference to positions. Some alignment-free methods are based on match length, which defines the distance between sequences based on the length of substring matches between two sequences. These include the shortest unique substring method (Haubold et al., 2005), ACS (Ulitsky et al. 2006), UA (Comin and Verzotto, 2012), and ALFRED (Thankachan et al. 2016). In addition, graphical representation was used to construct the probability distribution of a DNA sequence (Yu et al., 2011). The chaos game representation transforms the distribution of characters in a DNA sequence into the distribution of nodes in a graph (Hoang et al. 2016; Yin, 2017; Mendizabal-Ruiz et al., 2018). Many researchers considered extracting the position information of a k-mer (Huang and Wang, 2011; Ding et al., 2013; Tang et al., 2014). Ding et al. (2013) used the average interval distance of normalized k-mers to capture evolutionary information for sequence comparison. Tang et al. (2014) presented the average relative distance of normalized k-mers to improve the method of Ding et al. (2013). Ma et al. (2020) proposed the PWKmer method, which combines the k-mer counts and k-mer position distributions for phylogenetic analysis.

In this work, we propose a new alignment-free method which combines the position-weighted measure of k-mers proposed by Ma et al. (2020) and the information entropy of frequency of k-mers to obtain phylogenetic information for sequence comparison. It is named information-entropy position-weighted k-mer relative measure (IEPWRMkmer). To evaluate the performance of this method, we carry out phylogenetic analysis on two data sets used by other researchers.

Let S = s 1 s 2 ⋯ s L be a DNA sequence with length L, a 1 a 2 ⋯ a k is a k-mer, where a i ∈(A,T,C,G). If the k-mer a 1 a 2 ⋯ a k occurs in S, we denote by p a 1 a 2 ⋯ a k the vector composed of the positions of a 1 a 2 ⋯ a k in this given sequence and by p a 1 a 2 ⋯ a k   ( i ) its ith element. If the k-mer a 1 a 2 ⋯ a k does not occur in S,  we set   p a 1 a 2 ⋯ a k =(0). For example, for the DNA sequence GTA​ACC​TGA​ACG​TAC​TTG​GA with length 20, we list all 2-mer position vectors:

P _AA =(3,9); P _AC =(4,10,14); P _AG = (0); P _AT = (0); P _CA =(0); P _CC =(5); P _CG =(11); P _CT =(6,15); P _GA =(8,19); P _GC =(0); P _GG =(18); P _GT =(1,12); P _TA =(2,13); P _TC = 0; P _TG =(7,17); P _TT =(16).

In this example, the 2-mers AG, AT, CA, GC, and TC do not appear. For each k-mer, its position vector provides its position distribution information in the sequence. One can use the k-mer position vectors to reconstruct the DNA sequence (Ma et al., 2020).

Ma et al. (2020) defined the position-weighted measure D ( a 1 a 2 ⋯ a k ) of a 1 a 2 ⋯ a k based on its position in the sequence as D ( a 1 a 2 ⋯ a k ) = { ∑ i = 1 n p a 1 a 2 ⋯ a k ( i ) L ( L − k + 1 ) , n ≠ 0 , 0 , n = 0 , (1) where n is the length of the vector p a 1 a 2 ⋯ a k . Actually p a 1 a 2 ⋯ a k   ( i ) / L means the position weight of a 1 a 2 ⋯ a k in the given sequence with length L.

We denote by N the number of sequences in a dataset. In order to characterize the importance of k-mers in the whole dataset, we count the number m of the sequences that contain a k-mer a 1 a 2 ⋯ a k . Then the occurrence frequency F ( a 1 a 2 ⋯ a k ) of this k-mer in the whole dataset is defined as m/N. We introduce the Shannon entropy H( a 1 a 2 ⋯ a k ) of frequency F( a 1 a 2 ⋯ a k ) defined by Murray et al. (2017) as H ( a 1 a 2 ⋯ a k ) = − ( F ⁡ log 2 ( F ) + ( 1 − F ) log 2 ( 1 − F ) ) , (2) where F stands for F ( a 1 a 2 ⋯ a k ).

In this study, we aim to get more DNA phylogenetic information by combining the above two methods and defining E ( a 1 a 2 ⋯ a k ) = D ( a 1 a 2 ⋯ a k ) × H ( a 1 a 2 ⋯ a k ) (3)

Here, we regard Shannon entropy H ( a 1 a 2 ⋯ a k ) as another weight.

For a fixed K, there are 4 ^K k-mers. For each k-mer a 1 a 2 ⋯ a k , we can calculate the corresponding E ( a 1 a 2 ⋯ a k ) , then arrange 4 ^K of these E ( a 1 a 2 ⋯ a k ) to get a feature representation vector ( E 1 , E 2 , ⋯ , E 4 K ) according to the alphabet order of the 4 ^K k-mers for each genome.

For two given genome sequences A and B, we can obtain   E A = ( E 1 A , E 2 A , ⋯ , E 4 K A ) and E B = ( E 1 B , E 2 B , ⋯ , E 4 K B ) by the method. We use the Manhattan distance to calculate the pairwise distance between these two genome sequences: D ( A , B ) = ∑ i 4 K | ( E i A − E i B ) | .(4)

For a given dataset, we can derive a distance matrix by Eq. 4. This distance matrix contains the sequence similarity information. After obtaining the distance matrix, we insert it into the mega 7.0 software (Sudhir et al., 2016) and use Neighbor-Joining (NJ) program (Saitou et al. 1987) to construct the phylogenetic tree.

In this paper, a new alignment-free method is proposed for phylogenetic analysis and sequence comparison based on whole genome sequences. Our method combines the position-weighted measure of k-mers and the information entropy of frequency of k-mers. We used the Manhattan metric to measure the distance between a pair of sequences and the NJ method to construct the phylogenetic tree. In order to test the effectiveness and reliability of our method, we applied it on two datasets of 30 mammalian species and 44 HIV-1 genomes. The results demonstrated that the present method is efficient and reliable. A suitable K value is important to capture rich phylogenetic information of DNA sequences. In order to choose an optimal K value, we proposed a scoring measure based on the information entropy. The obtained results on two real datasets support that the method can capture the k-mer distribution information and is effective for whole genome sequence comparison and phylogenetic analysis.

Remark: The method of this paper is derived from the two studies Ma et al. (2020) and Murray et al. (2017). There are differences between this work and previous works: Tang et al. presented the average relative distance for normalized k-mers. PWKmer uses the counts and position distributions of k-mers to capture more evolutionary information. KWIP (Murray et al. 2017) uses information entropy to weight the inner product (Si ∗ Sj), while we use information entropy to weight the relative positions of k-mers. KWIP uses a kernel function to calculate the distance, while we use the Manhattan metric to calculate the pairwise distance between species. Here, we claimed that the results obtained by the IEPWRMkmer method are close to those by ClustalX and the IEPWRMkmer is superior to the other distance metrics. We used the phylogenetic tree constructed by ClustalX as the reference tree or standard tree, hence we cannot claim that our method is superior to the ClustalX method.