- Research
- Open Access
- Published:

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

*EURASIP Journal on Bioinformatics and Systems Biology*
**volume 2014**, Article number: 1 (2014)

## Abstract

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.

## 1. 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 [1–5]. 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.

Until now, there are many different graphical representation methods having been proposed to numerically characterize DNA sequences on the basis of different multiple-dimension spaces. For example, Liao et al. [6–9], Randic et al. [10–13], Guo et al. [14, 15], Qi et al. [16], Dai et al. [17, 18], and Dorota et al. [19] proposed different 2D graphical representation methods of DNA sequences, respectively. Liao et al. [20–23], Randic et al. [24, 25], Qi et al. [26], Yu et al. [27], and Aram et al. [28] proposed different 3D graphical representation methods of DNA sequences, respectively. Liao et al. [29], Tang et al. [30], and Chi et al. [31] proposed different 4D graphical representation methods of DNA sequences, respectively. In addition, Liao et al. [32] also proposed a kind of 5D representation method of DNA sequences and so on.

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.

## 2. 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 β-globin genes of human, chimpanzee, and opossum are shown in Figures 1, 2, and 3, respectively.

## 3. 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 *t*_{
i
} repeats in the sequence of triplets *G* = *t*_{1}, *t*_{2}, *t*_{3}…*t*_{
M
} for any *i* ∈ {1, 2,…, *M*}.

Let *T*_{1} = GCT, *T*_{2} = GCC, *T*_{3} = GCA, *T*_{4} = GCG, *T*_{5} = CGT, *T*_{6} = CGC, *T*_{7} = CGA, *T*_{8} = CGG, *T*_{9} = AGA, *T*_{10} = AGG, *T*_{11} = GAT, *T*_{12} = GAC, *T*_{13} = AAT, *T*_{14} = AAC, *T*_{15} = TGT, *T*_{16} = TGC, *T*_{17} = GAA, *T*_{18} = GAG, *T*_{19} = CAA, *T*_{20} = CAG, *T*_{21} = GGT, *T*_{22} = GGC, *T*_{23} = GGA, *T*_{24} = GGG, *T*_{25} = CAT, *T*_{26} = CAC, *T*_{27} = ATT, *T*_{28} = ATC, *T*_{29} = ATA, *T*_{30} = CTT *T*_{31} = CTC, *T*_{32} = CTA, *T*_{33} = CTG, *T*_{34} = TTA, *T*_{35} = TTG, *T*_{36} = AAA, *T*_{37} = AAG, *T*_{38} = TTT, *T*_{39} = TTC, *T*_{40} = CCT, *T*_{41} = CCC, *T*_{42} = CCA, *T*_{43} = CCG, *T*_{44} = TCT, *T*_{45} = TCC, *T*_{46} = TCA, *T*_{47} = TCG, *T*_{48} = AGT, *T*_{49} = AGC, *T*_{50} = ACT, *T*_{51} = ACC, *T*_{52} = ACA, *T*_{53} = ACG, *T*_{54} = TGG, *T*_{55} = TAT, *T*_{56} = TAC, *T*_{57} = GTT, *T*_{58} = GTC, *T*_{59} = GTA, *T*_{60} = GTG, *T*_{61} = ATG, *T*_{62} = TAA, *T*_{63} = TAG, and *T*_{64} = TGA.

Thereafter, according to Table 2, since there are a total of 64 triplets of DNA, then we can construct a set of 64 vectors {<*T*_{1}, *δ* (*T*_{1})>, <*T*_{2}, *δ* (*T*_{2})>,…, <*T*_{64}, *δ* (*T*_{64})>} for the given sequence of triplets *G* = *t*_{1}, *t*_{2}, *t*_{3}…*t*_{
M
} as follows: if *T*_{
i
} = *t*_{
j
} ∈ {*t*_{1}, *t*_{2}, *t*_{3},…*t*_{
M
}}, then *δ* (*T*_{
i
}) = *δ* (*t*_{
j
}), else *δ* (*T*_{
i
}) =0, for any *i* ∈ {1, 2,…, 64} and *j* ∈ {1, 2,…, *M*}.

For convenience, we call {<*T*_{1}, *δ* (*T*_{1})>, <*T*_{2}, *δ* (*T*_{2})>,…, <*T*_{64}, *δ* (*T*_{64})>} as the triplet-repeat model set of *G*.

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 β-globin 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 human-gorilla 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 human-opossum and human-lemur, which is obviously not reasonable also.

## 4. 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.

## References

- 1.
Chen W, Liao B, Liu Y, Zhu W, Su Z:

**A numerical representation of DNA sequences and its applications.***MATCH: Commun Math Comput Chem.*2008,**60:**291-300. - 2.
Jafarzadeh N, Iranmanesh A:

**A novel graphical and numerical representation for analyzing DNA sequences based on codons.***MATCH: Commun Math Comput Chem.*2012,**68:**611-620. - 3.
Liao B, Liao BY, Sun XM, Zeng QG:

**A novel method for similarity analysis and protein sub-cellular localization prediction.***Bioinformatics*2010,**26:**2678-2683. 10.1093/bioinformatics/btq521 - 4.
Qi XQ, Wu Q, Zhang Y, Fuller E, Zhang CQ:

**A novel model for DNA sequence similarity analysis based on graph theory.***J Evol Bioinform*2011,**7:**149-158. - 5.
Yu JF, Wang JH, Sun X:

**Analysis of similarities/dissimilarities of DNA sequences based on a novel graphical representation.***MATCH: Commun Math Comput Chem.*2010,**63:**493-512. - 6.
Li Y, Huang G, Liao B, Liu Z:

**H-L curve: a novel 2D graphical representation of protein sequences.***MATCH: Commun Math Comput Chem.*2009,**61:**519-532. - 7.
Liao B, Wang TM:

**New 2D graphical representation of DNA sequences.***J. Comput. Chem.*2004,**25:**1364-1368. 10.1002/jcc.20060 - 8.
Liao B, Xiang XY, Zhu W:

**Coronavirus phylogeny based on 2D graphical representation of DNA sequence.***J. Comput. Chem.*2006,**27:**1196-1202. 10.1002/jcc.20439 - 9.
Liu ZB, Liao B, Zhu W, Huang GH:

**A 2D graphical representation of DNA sequence based on dual nucleotides and its application.***Int. J. Quantum Chem.*2009,**109:**948-958. 10.1002/qua.21919 - 10.
Randic M, Vracko M, Zupan J, Novic M:

**Compact 2D graphical representation of DNA.***Chem. Phys. Lett.*2003,**373:**558-562. 10.1016/S0009-2614(03)00639-0 - 11.
Randic M, Vracko M, Lers N, Plavsic D:

**Analysis of similarity/dissimilarity of 2D graphical representation.***Chem. Phys. Lett.*2003,**371:**202-207. 10.1016/S0009-2614(03)00244-6 - 12.
Randic M, Vracko M, Lers N, Plavsic D:

**Novel 2-D graphical representation of DNA sequences and their numerical characterization.***Chem. Phys. Lett.*2003,**368:**1-6. 10.1016/S0009-2614(02)01784-0 - 13.
Randic M:

**Graphical representations of DNA as 2-D map.***Chem. Phys. Lett.*2004,**386:**468-471. 10.1016/j.cplett.2004.01.088 - 14.
Guo XF, Randic M, Basak SC:

**A novel 2-D graphical representation of DNA sequences of low degeneracy.***Chem. Phys. Lett.*2001,**350:**106-112. 10.1016/S0009-2614(01)01246-5 - 15.
Guo XF, Nandy A:

**Numerical characterization of DNA sequences in a 2-D graphical representation scheme of low degeneracy.***Chem. Phys. Lett.*2003,**369:**361-366. 10.1016/S0009-2614(02)02029-8 - 16.
Qi ZH, Qi XQ:

**Novel 2D graphical representation of DNA sequence based on dual nucleotides.***Chem Phys Lett.*2007,**440:**139-144. 10.1016/j.cplett.2007.03.107 - 17.
Dai Q, Xiu ZL, Wang TM:

**A novel 2D graphical representation of DNA sequences and its application.***J Mol Graph Model.*2006,**25:**340-344. 10.1016/j.jmgm.2005.12.004 - 18.
Liu XQ, Dai Q, Xiu ZL, Wang TM:

**PNN–curve: a new 2D graphical representation of DNA sequences and its application.***J. Theor. Biol.*2006,**243:**555-561. 10.1016/j.jtbi.2006.07.018 - 19.
Dorota BW, Timothy C, Piotr W:

**2D-dynamic representation of DNA sequences.***Chem. Phys. Lett.*2007,**442:**140-144. 10.1016/j.cplett.2007.05.050 - 20.
Yuan CX, Liao B, Wang TM:

**New 3D graphical representation of DNA sequences and their numerical characterization.***Chem. Phys. Lett.*2003,**379:**412-417. 10.1016/j.cplett.2003.07.023 - 21.
Liao B, Wang TM:

**3-D graphical representation of DNA sequences and their numerical characterization.***J. Mol. Struct. (THEOCHEM)*2004,**681:**209-212. 10.1016/j.theochem.2004.05.020 - 22.
Liao B, Wang TM:

**A 3D graphical representation of RNA secondary structure.***J Biomol Struct Dynam.*2004,**21:**827-832. 10.1080/07391102.2004.10506972 - 23.
Cao Z, Liao B, Li RF:

**A group of 3D graphical representation of DNA sequences based on dual nucleotides.***Int. J. Quantum Chem.*2008,**108:**1485-1490. 10.1002/qua.21698 - 24.
Randic M, Vracko M, Nandy A, Basak SC:

**On 3D graphical representation of DNA primary sequences and their numerical characterization.***J. Chem. Inf. Comput. Sci.*2000,**40:**1235-1244. 10.1021/ci000034q - 25.
Randic M, Zupan J, Novic M:

**On 3D graphical representation of proteomics maps and their numerical characterization.***J. Chem. Inf. Comput. Sci.*2001,**41:**1339-1344. 10.1021/ci0001684 - 26.
Qi XQ, Fan TR:

**PN-curve: a 3D graphical representation of DNA sequences and their numerical characterization.***Chem. Phys. Lett.*2007,**442:**434-440. 10.1016/j.cplett.2007.06.029 - 27.
Yu JF, Sun X, Wang JH:

**TN curve: a novel 3D graphical representation of DNA sequence based on trinucleotides and its applications.***J. Theor. Biol.*2009,**261:**459-468. 10.1016/j.jtbi.2009.08.005 - 28.
Aram V, Iranmanesh A:

**3D-dynamic representation of DNA sequences.***MATCH: Commun Math Comput Chem.*2012,**67:**809-816. - 29.
Liao B, Tan MS, Ding KQ:

**A 4D representation of DNA sequences and its application.***Chem. Phys. Lett.*2005,**402:**380-383. 10.1016/j.cplett.2004.12.062 - 30.
Tang XC, Zhou PP, Qiu WY:

**On the similarity/dissimilarity of DNA sequences based on 4D graphical representation.***Chin. Sci. Bull.*2010,**55:**701-704. 10.1007/s11434-010-0045-2 - 31.
Chi R, Ding KQ:

**Novel 4D numerical representation of DNA sequences.***Chem. Phys. Lett.*2005,**407:**63-67. 10.1016/j.cplett.2005.03.056 - 32.
Liao B, Xiang XY, Li RF, Zhu W:

**On the similarity of DNA primary sequences based on 5D representation.***J. Math. Chem.*2007,**42:**47-57. 10.1007/s10910-006-9091-z - 33.
He P, Wang J:

**Characteristic sequences for DNA primary sequence.***J. Chem. Inf. Comput. Sci.*2002,**42:**1080-1085. 10.1021/ci010131z

## Acknowledgements

This work is supported by the Chongqing Education Science Project of China in 2014, Chongqing “Twelfth Five Year plan” educational programming projects of China (2013-ZJ-077), program for university youth backbone teachers of Chongqing in 2014.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Competing interests

The authors declare that they have no competing interests.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

**Open Access**
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Zou, S., Wang, L. & Wang, J. A 2D graphical representation of the sequences of DNA based on triplets and its application.
*J Bioinform Sys Biology* **2014, **1 (2014). https://doi.org/10.1186/1687-4153-2014-1

Received:

Accepted:

Published:

### Keywords

- Graphical representation
- Similarities/dissimilarities analysis
- Triplet
- DNA sequence