key: cord-0921011-zocunore
authors: Liao, Bo; Liu, Yanshu; Li, Renfa; Zhu, Wen
title: Coronavirus phylogeny based on triplets of nucleic acids bases
date: 2006-04-15
journal: Chem Phys Lett
DOI: 10.1016/j.cplett.2006.01.030
sha: f19c52b0a6ded6d0c66c4049811963dd9cc7590e
doc_id: 921011
cord_uid: zocunore

We considered the fully overlapping triplets of nucleotide bases and proposed a 2D graphical representation of protein sequences consisting of 20 amino acids and a stop code. Based on this 2D graphical representation, we outlined a new approach to analyze the phylogenetic relationships of coronaviruses by constructing a covariance matrix. The evolutionary distances are obtained through measuring the differences among the two-dimensional curves.

Compilation of DNA primary sequence data continues unabated and tends to overwhelm us with voluminous outputs that increase daily. Comparison of primary sequences of different DNA strands remains one of the important aspect of the analysis of DNA data banks. Mathematical analysis of the large volume genomic DNA sequence data is one of the challenges for bio-scientists. There are three class methods for the analysis of DNA sequences: (i) Alignment [1, 2] . (ii) Matrices: (1) matrices in which an individual entry corresponds to an individual pair of bases [3, 6, 7] and (2) matrices in which entries summarize information of different X-Y pairs of bases [4, 5, 7] . (iii) Graphical representation: Graphical representation of DNA sequence provides a simple way of viewing, sorting and comparing various gene structures. Graphical techniques have emerged as a very powerful tool for the visualization and analysis of long DNA sequences. These techniques provide useful insights into local and global characteristics and the occurrences, variations and repetition of the nucleotides along a sequence which are not as easily obtainable by other meth-ods. In recent years several authors outlined different graphical representation of DNA sequences based on 2D, 3D or 4D [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] . Based on these graphical representation, several authors outlined some approaches to make comparison of DNA sequences [21] [22] [23] [24] [25] .

All this methods are based on the (four letter alphabet, A, C, G, and T standing for nucleotide bases adenine, cytosine, guanine, and thymine, respectively). We will change to consider the fully overlapping triplets of nucleotide bases. Consideration of triplets of nucleotide bases instead of individual nucleotide bases has several reasons and advantages. There are three of them: (i) The genetic code consists of triplets (codons) of DNA (or RNA in some virus) nucleotides. (ii) The second advantage is that one can easily find the open reading frame as the longest sequence of triplets that contains no stop codons when read in a single reading frame. (iii) The computation will become more simple.

In this Letter, we proposed a 2D graphical representation of the protein sequences consisting of 20 amino acids and a stop code. Based on this 2D graphical representation, we outlined a new approach to analyze the phylogenetic relationships of coronaviruses. The evolutionary distances are obtained through measuring the differences among the two-dimensional curves. Unlike most existing phylogeny construction methods [26] [27] [28] [29] [30] [31] , the proposed method does not require multiple alignment. 

As is known, all of the 64 triplets of nucleotide bases correspond 20 amino acids and a stop code. There are three reading frame start at position 1, 2 and 3, respectively. Using the translate tool, we can obtain three protein sequences consisting of 20 amino acids and a stop code. The 20 amino acids found in proteins can be grouped according to the chemistry of their R groups as in [32] : amino acids A,V,F,P,M,I,L belong to the hydrophobic chemical group; amino acids D,E,K,R belong to charged chemical group; amino acids S,T,Y,H,C,N,Q,W belong to polar chemical group; amino acid belong to glycine chemical group. Then for any DNA sequence, we will transform it into three new sequences defined over alphabet fH ; C; P ; Gg. The rule is as follows:

As shown in Fig. 1 , we construct a pyrimidine-purine graph on two quadrants of the cartesian coordinate system, with pyrimidines (P and C) in the first quadrant and purines (H and G) in the fourth quadrant. The unit vectors representing four alphabets H ; G; C and P are as follows:

where m is a real number and m 6 ¼ ffiffi ffi n p , n is a positive real number but not a perfect square number. So that we will reduce a DNA sequence into a series of nodes P 0 ,P 1 ,P 2 , . . . ,P ºN/3ß , whose coordinates x i , y i (i = 0,1, 2, . . . , ºN/3ß, where N is the length of the DNA sequence being studied) satisfy

h i ; c i ; g i and p i satisfy 

We called the corresponding plot set be characteristic plot set. The curve connected all plots of the characteristic plot set in turn is called characteristic curve, which is determined by m, n, that satisfy above mentioned condition. In Figs Proof. Using the translate tool, one can obtain three protein sequences consisting of 20 amino acids and a stop code corresponding three reading frame start at position 1, 2 and 3. In a single reading frame, let (x i , y i ) be the coordinates of the ith amino acid of protein sequence, then we have 

Obviously, x i and y i are irrational numbers of form sm þ k ffiffi ffi n p , where s and k are integers. We suppose

So, for given x-projection and y-projection of any point P = (x, y) on the sequence, after uniquely determining s x ,k x ,s y ,k y from x and y, the number A p ,V p ,F p ,P p , F,P,M,I,L,D,E,K,R,S,T,Y ,H,C,N,Q,W,G and À(or stop code) from the beginning of the sequence to the point P can be found by solving linear system (2) and (4).

The vector pointing to the point P i from the origin O is denoted by r i . The component of r i , i.e. x i and y i are calculated by Eqs. (1) and (2). Let Dr i = r i À r i À 1 , then we have Property 2. 1) and (2) . For example, when the ith residue is A, we find Dx i = m and Dy i ¼ À ffiffi ffi n p . This result is independent of the conformation state of the (i À 1)th residue. The two numbers ðm; À ffiffi ffi n p Þ are called the direction of Dr i . The direction number and the length of Dr i for each possible residue type at the ith position are summarized. h Property 3. There is no circuit or degeneracy in our twodimensional graphical representation.

Proof. We assume that: (1) the number of amino acid forming a circuit is l; (2) the number of A,V,F,P,M,I,L,-D,E,K,R,S, T,Y,H,C,N,Q,W,G and À(or stop code) in a circuit is a 0 ,v 0 ,f 0 ,p 0 ,m 0 ,i 0 ,l 0 ,d 0 ,e 0 ,k 0 ,r 0 ,s 0 ,t 0 ,y 0 ,h 0 ,c 0 ,n 0 ,q 0 ,w 0 ,g 0 and d 0 , respectively. So

n 0 N,q 0 Q,w 0 W,g 0 G and d 0 À(or stop code) form a circuit, the following equation holds: i.e., h 0 m þ g 0 ffiffi ffi n p þ c 0 ffiffi ffi n p þ Proof. usually the sequence is expressed in the order from 5 0 to 3 0 . Suppose that the 2D representation for protein sequence is described by (x i , y i ),i = 0,1, 2, . . . , N. Suppose again that the 2D representation for the reverse sequence, i.e, the same sequence but from 3 0 to 5 0 is described by ðx i ;ŷ i Þ, we find

h

For any DNA sequence, we have three translating protein sequences. For any protein sequence, we have a set of points (x i , y i ),i = 1,2,3, . . . ,N, where N is the length of the sequence. The coordinates of the geometrical center of the points, denoted by x 0 and y 0 , may be calculated as follows:

The element of covariance matrix CM of the points are defined:

(See Table 1 )The above four numbers give a quantitative description of a set of point (x i , y i ),i = 1,2,. . . , N, scattering in a two-dimensional space. Obviously, the matrix is a real symmetric 2 · 2 one. There is a leading eigenvalue for a matrix CM. So that there are three geometrical centers and three leading eigenvalue corresponding a DNA sequence.

In Table 2 , we list the geometrical centers ðx 0 k ; y 0 k Þ; k ¼ 1; 2; 3 and leading eigenvalues belonging to 24 species with parameter m ¼ 1 Table 3 ).

In order to facilitate the quantitative comparison of different species in terms of their collective parameters, we introduce a distance scale as defined below. Suppose that there are two species i and j, the parameters are

are the three leading eigenvalues of matrix CM i corresponding to species i. The distance d ij between the two points is where d ij denotes the distance between the geometric centers of the ith and the jth genomes, and M is the total number of all genomes (M = 24, here). Then we obtain a real M · M symmetric matrix whose elements are d ij . Accordingly, a real symmetric M · M matrix D ij is obtained and used to reflect the evolutionary distance between the species i and j. The clustering tree is constructed using the UPGMA method in PHYLIP package (http://evolution.genetics.washington.edu/phylip.html). The final phylogenetic tree is drawn using the DRAWGRAM program in the PHYLIP package. In Fig. 5 , we present the phylogenetic tree belonging to 24 species.

We made a analysis of DNA sequences by considering the fully overlapping triplets of nucleotide bases. The presented graphical representation can be recaptured mathe- matically without loss of textual information. And our representation provides a direct plotting method to denote DNA sequences without degeneracy. Most existing approaches for phylogenetic inference use multiple alignment of sequences and assume some sort of an evolutionary model. The multiple alignment strategy does not work for all types of data, e.g., whole genome phylogeny, and the evolutionary models may not always be correct. The current two-dimensional graphical representation of DNA sequences provides different approach for constructing phylogenetic tree. Unlike most existing phylogeny construction methods, the proposed method does not require multiple alignment. Also, both computational scientists and molecular biologists can use it to analysis protein sequences efficiently. We can obtain some graphical representation of protein sequence based on 2D, 3D and 4D using the following transform: a i ! h i ; g i ! g i ; c i ! c i ; t i ! P i . h i ; c i ; g i and p i satisfy Eq. (2). a i ,c i ,g i and t i are the cumulative occurrence numbers of A, C, G and T, respectively, in the subsequence from the 1st base to the ith base in the sequence.

String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison

Ho Yee-kin

Molecular Evolution Phylogeny

This work is supported in part by the China Postdoctoral Science Foundation and the National Natural Science Foundation of Hunan University.

A m À ffiffi ffi n p m 2 + n D ffiffi ffi n p m m 2 + n S ffiffi ffi n p Àm Table 3 The geometric centers and three leading eigenvalues for each of the 24 coronavirus genomes