A 2D graphical representation of the sequences of DNA based on triplets and its application

In this paper, we first present a new concept of ‘weight’ for 64 triplets and define a different weight for each kind of triplet. Then, we give a novel 2D graphical representation for DNA sequences, which can transform a DNA sequence into a plot set to facilitate quantitative comparisons of DNA sequences. Thereafter, associating with a newly designed measure of similarity, we introduce a novel approach to make similarities/dissimilarities analysis of DNA sequences. Finally, the applications in similarities/dissimilarities analysis of the complete coding sequences of β-globin genes of 11 species illustrate the utilities of our newly proposed method.


Introduction
In the recent years, an exponential growth of sequence data in DNA databases has been observed by biologists; the importance of understanding genetic sequences coupled with the difficulty of working with such immense volumes of DNA sequence data underscores the urgent need for supportive visual tools. Recently, graphical representation is well regarded which can offer visual inspection of data and provide a simple way to facilitate the similarity analysis and comparison of DNA sequences [1][2][3][4][5]. Because of its convenience and excellent maneuverability, currently, all kinds of methods based on graphical representation have been extensively applied in relevant realms of bioinformatics.
In these approaches mentioned above, most of them adopt the leading eigenvalues of some matrices, such as L/L matrices, M/M matrices, E matrices, covariance matrices, and D/D matrices, to weigh the similarities/ dissimilarities among the complete coding sequences of β-globin genes of different species. Because the matrix computation is needed to obtain the leading eigenvalues, these methods are usually computationally expensive for long DNA sequences. Furthermore, in some of these approaches, their results of similarities/dissimilarities analysis are not quite reasonable, and there are some results that do not accord with the fact [7,9].
To degrade the computational complexity and obtain more reasonable results of similarities/dissimilarities analysis of DNA sequences, in this article, we propose a new 2D graphical representation of DNA sequences based on triplets, in which, we present a new concept of 'weight' for 64 triplets and a new concept of 'weight deviation' to weigh the similarities/dissimilarities among the complete coding sequences of β-globin genes of different species. Compared with some existing graphical representations of the DNA sequences, our new scheme has the following advantages: (1) no matrix computation is needed, and (2) it can characterize the graphical representations for DNA sequences exactly and obtain reasonable results of similarities/dissimilarities analysis of DNA sequences.

Proposed 2D graphical representation of DNA sequence
Codon is a specific sequence of three adjacent nucleotides on the mRNA that specifies the genetic code information for synthesizing a particular amino acid. As illustrated in Table 1, there are total 20 amino acids and 64 codons in the natural world, and each of these codons has a specific meaning in protein synthesis: 64 codons represent amino acids and the other 3 codons cause the termination of protein synthesis.
For the 64 codons illustrated in Table 1, their corresponding triplets of DNA are illustrated in Table 2.
Based on the above 64 triplets of DNA illustrated in Table 2, we define a new mapping Ψ to map each of these triplets into a different weight. Obviously, the mapping Ψ shall satisfy the following rule: for any two pairs of triplets (X 1 , Y 1 ) and (X 2 , Y 2 ), where X 1 , Y 1 , X 2 , and Y 2 are all triplets, if the corresponding codons of X 1 and Y 1 code the same amino acid but the corresponding codons of X 2 and Y 2 code two different amino acids, then there shall be |Ψ (X 1 ) − Ψ (Y 1 )| < |Ψ (X 2 ) − Ψ (Y 2 )|. So, according to the above rule and for the sake of convenience, weights consist of amino acid and codon. Amino acid is the integer part of weight, and codon is the fractional part of weight. Alanine is defined as 1, arginine is defined as 2, and the rest can be done in the same manner. Codons of every amino acid are reordered, so the first codon of alanine's (GCT) weight value is 1.1. We design the detailed mapping rules of Ψ as illustrated in Table 3.
For example, from Table 3, we will have Ψ (GCT) = 1.1, Ψ (GCC) = 1.2, Ψ (ATG) = 20.1, etc., and in addition, we can propose a novel 2D graphical representation of DNA sequences as follows: Let G = g 1 , g 2 , g 3 …g N be an arbitrary DNA primary sequence, where g i ∈ {A, T, G, C} for any i ∈ {1, 2,…, N}, and then, we can transform G into a sequence of triplets such as G = t 1 , t 2 , t 3 …t M , where M = [N/3] and t i is a triplet of DNA for any i ∈ {1, 2,…, M}. Thereafter, we can define a new mapping Θ to map G into a plot set as illustrated in the formula (1).
As for the complete coding sequences of β-globin genes of 11 species illustrated in the Table 4, each of them can be mapped into a plot set by using the new given mapping Θ, and the 2D graphical representations corresponding to the complete coding sequences of βglobin genes of human, chimpanzee, and opossum are shown in Figures 1, 2, and 3, respectively.

Similarity analysis of DNA sequence
Let G = g 1 , g 2 , g 3 …g N be an arbitrary complete coding sequence, where g i ∈ {A, T, G, C} for any i ∈ {1, 2,…, N}, and G = t 1 , t 2 , t 3 …t M be its corresponding sequence of triplets, where M = [N/3] and t i is a triplet of DNA for any i ∈ {1, 2,…, M}. Then, we define a function δ and let δ (t i ) represent the total number of times that the triplet For any two given complete coding sequences A and B, suppose that their triplet-repeat model sets are {<T 1 , X 1 >, <T 2 , X 2 >,…, <T 64 , X 64 >} and {<T 1 , Y 1 >, <T 2 , Y 2 >,…, <T 64 , Y 64 >}, respectively. Then, on the basis of the 2D graphical representation given in the previous Section 2, we can define the weight deviation between the two DNA sequences A and B as the following formula (2) to measure the similarity between A and B.
Obviously, the above formula (2) satisfies the fact that the smaller the weight deviation between the two DNA sequences A and B, the higher the degree of similarity of A and B. According to formula (2), the detailed similarity/dissimilarity matrix obtained for the coding sequences listed in Table 4 is illustrated in Table 5. Basing on the similarity matrix (Table 5) constructs a phylogenetic tree, which is shown in Figure 4.
Observing Table 5, it is easy to find out that human, gorilla, and chimpanzee are most similar to each other, and the pairs like gorilla-chimpanzee (with weight deviation of 1.1266), human-gorilla (with weight deviation of 4.3359), and human-chimpanzee (with weight deviation of 5.2500) are the most similar species pairs, but Gallus and opossum are the most dissimilar to the others (with weight deviation bigger than 11). It is consistent with the fact that Gallus is not a mammal, whereas the others  are mammals, and opossum is the most remote species from the remaining mammals. Similar results have been obtained in other papers by different approaches [2,5,7,9,33]. For testing the validity of our method, the existing results of the examination of the degree of similarity/ dissimilarity of the coding sequences of β-globin genes of several species with the coding sequence of the human βglobin gene by means of approaches using alternative DNA sequence descriptors [2,5,7,9] are listed in Table 6 for comparison.
From Table 6, we can find that the pairs like humangorilla and human-chimpanzee are the two most similar species pairs when adopting (A) the method of our work, (B) the method of [2], (C) the method of [5], and (D) the method of [7], which is in accordance with the fact that gorilla and chimpanzee are the two most closest species of human, but when adopting (E) the method of [9], the most similar species pair is human-goat, which is obviously not correct. In addition, the pairs like human-Gallus and human-opossum are the two most dissimilar species pairs when adopting (A) the method of our work, (C) the method of [5], and (E) the method of [9], which is in accordance with the fact that Gallus is not a mammal, whereas the others are mammals, and opossum is the most remote species from the remaining mammals. However, when adopting (D) the method of [7], the two most dissimilar species pairs are humanopossum and human-lemur, which is obviously not reasonable also.  1  8  15  22  29  36  43  50  57  64  71  78  85  92  99  106  113  120  127  134 141 148 Figure 1 The 2D graphical representations of the complete coding sequences of β-globin genes of human.   1  8  15  22  29  36  43  50  57  64  71  78  85  92  99  106  113  120  127 134 141 148 Figure 3 The 2D graphical representations of the complete coding sequences of β-globin genes of opossum. Gallus 0 Figure 4 Phylogenetic tree based on the similarity matrix (Table 5).

Conclusion
In this paper, we propose a new 2D graphical representation for DNA sequences based on triplets, and associating with a newly introduced concept of weight of triplets and a newly designed measure of similarity named weight deviation, we propose a new method to make similarity analysis of DNA sequences, in which no matrix computation is needed and reasonable and useful approaches for both computational scientists and molecular biologists to effectively analyze DNA sequences can be provided at the same time.