# DV-Curve Representation of Protein Sequences and Its Application

Authors: Wei Deng; Yihui Luan
Journal: Computational and Mathematical Methods in Medicine
 (2014)

---

## Abstract

Based on the detailed hydrophobic-hydrophilic(HP) model of amino acids, we propose dual-vector curve (DV-curve) representation of protein sequences, which uses two vectors to represent one alphabet of protein sequences. This graphical representation not only avoids degeneracy, but also has good visualization no matter how long these sequences are, and can reflect the length of protein sequence. Then we transform the 2D-graphical representation into a numerical characterization that can facilitate quantitative comparison of protein sequences. The utility of this approach is illustrated by two examples: one is similarity/dissimilarity comparison among different ND6 protein sequences based on their DV-curve figures the other is the phylogenetic analysis among coronaviruses based on their spike proteins.


---

## Body

## 1. Introduction

The graphical representation method has become very common to analyze the huge amount of gene data. Generally, with this method we can first observe visual qualitative inspection in order to recognize major differences among similar gene sequences and further draw some mathematical characterizations of sequences to analyze their similarity/dissimilarity and evolutionary homology.Letter sequence representation (LSR) of DNA sequences represents each base by a letter of four different letters such as A, T, G, and C. DNA sequences can be represented in different dimension spaces. For example, G-curve and H-curve [1] were first proposed by Hamori and Ruskin before thirty years. Later, Gates [2] established a 2D graphical representation that was simpler than H curve. However, Gate’s graphical representation has high degeneracy because of some circuits appearing in its curve. Several researchers in their recent studies have outlined different kinds of DNA sequences graphical representation based on 2D [3–11], 3D [12–15], 4D [16], 5D [17], and 6D [18]. Among these methods, we here stress DV-curve representation which was proposed by Zhang [10]. DV-curve uses two vectors to represent one alphabet of DNA sequences and avoids degeneracy and loss of information. Furthermore, DV-curve has good visualization no matter how long these sequences are and can reflect the length of the DNA sequence.LSR of protein sequences represents each amino acid by a letter of twenty different letters such as A, R, N, D, C, Q, E, G, H, I, L, K, M, F, P, S, T, W, Y, and V. Although protein sequences and DNA sequences belong to symbolic sequences, the methods for the graphical representation of protein sequences are relatively less popular, compared with DNA sequences. The key reason is that the extension of DNA graphical representation to protein sequences enormously increases the number of possible alternative assignments for these 20 amino acids. The amino acid sequence is the key to discover protein structure and function in the cell, so analysis of amino acid sequences is a very important part of postgenomic studies. The graphical representation study of protein sequences emerged very recently. The first visualization protein model was proposed by Randić et al. until 2004 [19]. Some researchers have studied on graphical representation of protein sequences from different perspectives [20–29].In this paper, we introduce DV-curve graphical representation of protein sequences based on the detailed hydrophobic-hydrophilic (HP) model of amino acids. According to the important hydropathy, this approach is accompanied by a relatively small number of arbitrary choices associated with the graphical representation of proteins. Also, this representation has relatively good visualization effect to describe protein sequences in a perceivable way. As its application, we analyze the similarity/dissimilarity among some ND6 sequences and construct the phylogenetic tree of 35 coronavirus spike proteins.


## 2. DV-Curve Representation of Protein Sequences

### 2.1. Classification of Protein Sequences

The amino acid sequence is closely related to biological function. The closer the genetic relationship is, the smaller the difference in amino acid composition between them will be. Over the past thirty years, the characteristics of protein sequences have been studied by establishing different classified models [21–24, 26, 27]. A well-known model of protein sequences is the hydrophobic (H or nonpolar)-hydrophilic (P or polar), that is, the HP model may be too simple and lacks enough consideration on the heterogeneity and the complexity of the natural set of residues [30]. Based on Brown’s work [31], 20 different kinds of amino acids are divided into four groups: nonpolar (np), negative polar (nep), uncharged polar (up), and positive polar (pp). This is called the detailed HP model, which can provide more information than the original HP model.For a given protein sequenceS
=
S
1
S
2
⋯
S
n with length n, where S
i is the letter in the ith position among the protein sequence (i
=
1,2
,
…
,
n), we define a primary protein sequence as a symbolic sequence which includes four letters according to the following rule:
(1)
b
i
=
{
B
1
,
if
S
i
∈
np
,
B
2
,
if
S
i
∈
nep
,
B
3
,
if
S
i
∈
up
,
B
4
,
if
S
i
∈
pp
.Sob
i is the substitution for S
i, and then we obtain a sequence G
(
s
)
=
b
1
b
2
⋯
b
n. Here b
i is a letter of the alphabet B
1
,
B
2
,
B
3
,
B
4. For example, for a given protein primary sequence S
=
W
T
F
E
S
R
N
D
P
A
K, we can transform it into a new sequence according to the above rule, G
(
S
)
=
B
1
B
3
B
1
B
2
B
3
B
4
B
3
B
2
B
1
B
1
B
4. Via comparison of the reduced sequence, it will be easier to understand the biological function of various kinds of amino acid residues.


### 2.2. Graphical Representation of Protein Sequences

In this section, we will construct DV-curve representation of protein sequence. Given any protein primary sequence with lengthn, we can transform it into a new sequence composed of a character set of B
1
,
B
2
,
B
3
,
B
4. As shown in Figure 1, these alphabets are assigned, respectively, by consecutive vectors as follows:
(2)
B
1
⟹
(
1,1
)
,
(
1,1
)
B
2
⟹
(
1,1
)
,
(
1
,
-
1
)
B
3
⟹
(
1
,
-
1
)
,
(
1,1
)
B
4
⟹
(
1
,
-
1
)
,
(
1
,
-
1
)
.The representation of four alphabets of DV-curve: (a)B
1, (b) B
2, (c) B
3, and (d) B
4.
(a)
(b)
(c)
(d)We connect adjacent dots with lines and then obtain a dual-vector curve form. This process is shown in Figure2.Figure 2
The DV-curve of sequence “WTFESR.”Based on the construction of DV-curve, we obtain two mathematical models, respectively. One is “from protein sequence to DV-curve,” and the other is “from DV-curve to protein sequence.” Firstly, we give some common symbols and variables.(
1
) According to the classification rule, we describe a protein sequence as G
(
S
)
=
b
1
b
2
b
3
⋯
b
n, where b
i
∈
{
B
1
,
B
2
,
B
3
,
B
4
} with length n. It means that the protein sequence S is connected by these alphabets. (
2
)  (
x
i
,
y
i
) is the coordinate of the ith point of DV-curve, and (
x
0
,
y
0
)
=
(
0,0
) is the start point.Model One. Given a primary protein sequence, we can draw its DV-curve:
(3)
x
2
i
-
1
=
2
i
-
1
,
i
=
1
,
2
,
…
,
n
,
x
2
i
=
2
i
,
i
=
1
,
2
,
…
,
n
,
y
2
i
-
1
=
{
y
2
i
-
2
+
1
,
if
b
i
=
B
1
or
B
2
,
y
2
i
-
2
-
1
,
if
b
i
=
B
3
or
B
4
,
y
2
i
=
{
y
2
i
-
1
+
1
,
if
b
i
=
B
1
or
B
3
,
y
2
i
-
1
-
1
,
if
b
i
=
B
2
or
B
4
.According to the above four formulas, the coordinate of each point(
x
i
,
y
i
) can be calculated. Then we connect all the points with beelines, and the DV-curve is obtained.Model Two. Given a DV-curve, we can also obtain the coarse-grained description of the protein sequence based on the detailed HP-model:
(4)
G
(
S
i
)
=
{
B
1
,
if
y
2
i
-
1
-
y
2
i
-
2
=
1
,
y
2
i
-
y
2
i
-
1
=
1
,
B
2
,
if
y
2
i
-
1
-
y
2
i
-
2
=
1
,
y
2
i
-
y
2
i
-
1
=
-
1
,
B
3
,
if
y
2
i
-
1
-
y
2
i
-
2
=
-
1
,
y
2
i
-
y
2
i
-
1
=
1
,
B
4
,
if
y
2
i
-
1
-
y
2
i
-
2
=
-
1
,
y
2
i
-
y
2
i
-
1
=
-
1
.
Here i
=
1,2
,
3
,
…
,
n. If each point (
x
i
,
y
i
) of DV-curve is given in this model, we can get each B
i according to the above formulas. So the simplified protein sequence G
(
S
)
=
b
1
b
2
⋯
b
n can be recovered; here b
i
∈
{
B
1
,
B
2
,
B
3
,
B
4
} with length n.


## 2.1. Classification of Protein Sequences

The amino acid sequence is closely related to biological function. The closer the genetic relationship is, the smaller the difference in amino acid composition between them will be. Over the past thirty years, the characteristics of protein sequences have been studied by establishing different classified models [21–24, 26, 27]. A well-known model of protein sequences is the hydrophobic (H or nonpolar)-hydrophilic (P or polar), that is, the HP model may be too simple and lacks enough consideration on the heterogeneity and the complexity of the natural set of residues [30]. Based on Brown’s work [31], 20 different kinds of amino acids are divided into four groups: nonpolar (np), negative polar (nep), uncharged polar (up), and positive polar (pp). This is called the detailed HP model, which can provide more information than the original HP model.For a given protein sequenceS
=
S
1
S
2
⋯
S
n with length n, where S
i is the letter in the ith position among the protein sequence (i
=
1,2
,
…
,
n), we define a primary protein sequence as a symbolic sequence which includes four letters according to the following rule:
(1)
b
i
=
{
B
1
,
if
S
i
∈
np
,
B
2
,
if
S
i
∈
nep
,
B
3
,
if
S
i
∈
up
,
B
4
,
if
S
i
∈
pp
.Sob
i is the substitution for S
i, and then we obtain a sequence G
(
s
)
=
b
1
b
2
⋯
b
n. Here b
i is a letter of the alphabet B
1
,
B
2
,
B
3
,
B
4. For example, for a given protein primary sequence S
=
W
T
F
E
S
R
N
D
P
A
K, we can transform it into a new sequence according to the above rule, G
(
S
)
=
B
1
B
3
B
1
B
2
B
3
B
4
B
3
B
2
B
1
B
1
B
4. Via comparison of the reduced sequence, it will be easier to understand the biological function of various kinds of amino acid residues.


## 2.2. Graphical Representation of Protein Sequences

In this section, we will construct DV-curve representation of protein sequence. Given any protein primary sequence with lengthn, we can transform it into a new sequence composed of a character set of B
1
,
B
2
,
B
3
,
B
4. As shown in Figure 1, these alphabets are assigned, respectively, by consecutive vectors as follows:
(2)
B
1
⟹
(
1,1
)
,
(
1,1
)
B
2
⟹
(
1,1
)
,
(
1
,
-
1
)
B
3
⟹
(
1
,
-
1
)
,
(
1,1
)
B
4
⟹
(
1
,
-
1
)
,
(
1
,
-
1
)
.The representation of four alphabets of DV-curve: (a)B
1, (b) B
2, (c) B
3, and (d) B
4.
(a)
(b)
(c)
(d)We connect adjacent dots with lines and then obtain a dual-vector curve form. This process is shown in Figure2.Figure 2
The DV-curve of sequence “WTFESR.”Based on the construction of DV-curve, we obtain two mathematical models, respectively. One is “from protein sequence to DV-curve,” and the other is “from DV-curve to protein sequence.” Firstly, we give some common symbols and variables.(
1
) According to the classification rule, we describe a protein sequence as G
(
S
)
=
b
1
b
2
b
3
⋯
b
n, where b
i
∈
{
B
1
,
B
2
,
B
3
,
B
4
} with length n. It means that the protein sequence S is connected by these alphabets. (
2
)  (
x
i
,
y
i
) is the coordinate of the ith point of DV-curve, and (
x
0
,
y
0
)
=
(
0,0
) is the start point.Model One. Given a primary protein sequence, we can draw its DV-curve:
(3)
x
2
i
-
1
=
2
i
-
1
,
i
=
1
,
2
,
…
,
n
,
x
2
i
=
2
i
,
i
=
1
,
2
,
…
,
n
,
y
2
i
-
1
=
{
y
2
i
-
2
+
1
,
if
b
i
=
B
1
or
B
2
,
y
2
i
-
2
-
1
,
if
b
i
=
B
3
or
B
4
,
y
2
i
=
{
y
2
i
-
1
+
1
,
if
b
i
=
B
1
or
B
3
,
y
2
i
-
1
-
1
,
if
b
i
=
B
2
or
B
4
.According to the above four formulas, the coordinate of each point(
x
i
,
y
i
) can be calculated. Then we connect all the points with beelines, and the DV-curve is obtained.Model Two. Given a DV-curve, we can also obtain the coarse-grained description of the protein sequence based on the detailed HP-model:
(4)
G
(
S
i
)
=
{
B
1
,
if
y
2
i
-
1
-
y
2
i
-
2
=
1
,
y
2
i
-
y
2
i
-
1
=
1
,
B
2
,
if
y
2
i
-
1
-
y
2
i
-
2
=
1
,
y
2
i
-
y
2
i
-
1
=
-
1
,
B
3
,
if
y
2
i
-
1
-
y
2
i
-
2
=
-
1
,
y
2
i
-
y
2
i
-
1
=
1
,
B
4
,
if
y
2
i
-
1
-
y
2
i
-
2
=
-
1
,
y
2
i
-
y
2
i
-
1
=
-
1
.
Here i
=
1,2
,
3
,
…
,
n. If each point (
x
i
,
y
i
) of DV-curve is given in this model, we can get each B
i according to the above formulas. So the simplified protein sequence G
(
S
)
=
b
1
b
2
⋯
b
n can be recovered; here b
i
∈
{
B
1
,
B
2
,
B
3
,
B
4
} with length n.


## 3. Numerical Characterization of Protein Sequences

In order to facilitate quantitative comparisons of sequences, we will give numerical characterization of graphical curve as the descriptor. In general, we transform the graphical representation into a mathematical object like a matrix in order to draw some invariants. The frequently used matrices includeE matrix, M matrix, L matrix, and L
k matrix proposed by Randić et al. [6, 8, 32–34]. Of course, there are some other matrix invariants such as the average matrix element, the average row sum, the Wiener number, and the ALE-index et al. These methods were used widely and proved to be useful. Here, we use the C
M
x
y as an alternative sequence invariant proposed by Liao et al. [35]:
(5)
(
x
c
,
y
c
)
=
(
1
2
n
+
1
∑
i
=
0
2
n
‍
x
i
,
1
2
n
+
1
∑
i
=
0
2
n
‍
y
i
)
,
C
M
x
y
=
1
2
n
+
1
∑
i
=
0
2
n
‍
(
x
i
-
x
c
)
(
y
i
-
y
c
)
.Obviously, this index is relatively simple for calculation so that this index can provide some convenience for long sequences.If we adjust the order ofB
1
,
B
2
,
B
3
,
B
4 corresponding to basic dual vectors, we can get another curve. So for a given sequence, we can get 4
!
=
24 different DV-curves totally. Therefore, a protein primary sequence can be characterized by a 24-component vector as follows: v
→
=
[
C
M
1
x
y
,
C
M
2
x
y
,
…
,
C
M
2
4
x
y
]. Based on the vectors, we can compare different protein sequences. Generally speaking, we can obtain the similarities of the two vectors by calculating Euclidean distance. If two sequences are similar, the distance between two corresponding points should be small. Given two species i and j, the corresponding vectors are v
i
→
=
[
C
M
i
1
x
y
,
C
M
i
2
x
y
,
…
,
C
M
i
2
4
x
y
] and v
j
→
=
[
C
M
j
1
x
y
,
C
M
j
2
x
y
,
…
,
C
M
j
2
4
x
y
], respectively; then we have d
(
v
i
→
,
v
j
→
)
=
∑
k
=
1
24
(
C
M
i
k
x
y
-
C
M
j
k
x
y
)
2.


## 4. Application

The comparison on biology sequences is one of the most important parts in bioinformatics when analyzing similarities of function and properties. In this section, we will give two main applications of this new graphical representation. One is similarity analysis based on visual graphics. Generally, similarity analysis can be divided into two types of methodologies to conduct the comparison: sequence alignment and sequence descriptors comparison. When recognizing figures, our brain is more helpful for similarity analysis in multiple sequences. So it is desirable to propose similarity analysis by inspecting the DV-curve of protein. The other is evolutionary homology analysis based on the numerical characterization of DV-curve, and we construct a 24-component vector to characterize any protein sequence. As further work, the phylogenetic tree of 35 coronavirus spike proteins is constructed.


### 4.1. Similarity Analysis Based on Visual Inspection of the Protein DV-Curve Graphs

Since Smith and Waterman developed a dynamic programming algorithm in 1981, many alignment algorithms identifying whether two biological sequences are similar to each other have been studied. These methods are proved to be efficient. However, multiple sequence alignment (MSA) of several hundred sequences has always produced a bottleneck.In 1994, MSA was proved to be an NP-complete problem by Wang and Jiang [36]. Moreover, most experts think that it is impossible until now to build a deterministic polynomial algorithm to handle an NP-complete problem. It needs to exhaust almost billions or trillions of years. Except long computational time, there also exists possible bias of multiple sequence alignments for multiple occurrences of highly similar sequence [37].However, our brain is much more powerful than computer when recognizing different figures. So it can help us to analyze the similarity in multiple sequences. If we can provide a simple, intuitional, clear, and nondegenerate 2D graphical representation of protein sequences, molecular biologists may easily find out which sequence is most similar or dissimilar to the given target sequence. And next they can use alignment algorithms for further confirmation.According to our proposed definition of protein DV-curve, we can draw the curves of some ND6 (NADH dehydrogenase subunit 6) proteins in order to conveniently compare them. Protein sequences that are used to prove our approach were downloaded from GenBank: human(
Y
P
_
003024037.1
), gorilla (
N
P
_
008223
), chimpanzee (
N
P
_
008197
), wallaroo (
N
P
_
007405
), harbor seal (H. seal) (
N
P
_
006939
), gray seal (G. seal) (
N
P
_
007080
), rat (
A
P
_
004903
), and mouse (
N
P
_
904339
), and the same data set was used in [26, 27].In Figure3, it is evident that protein graph of wallaroo is obviously different from the other species because it is the most remote species from the remaining mammals. Furthermore, we can see human and chimpanzee have similar curves, harbor seal and gray seal’s curves are almost identical, and two curves of rat and mouse are very similar. All these results not only are consistent with the conclusions drawn by Smith-Waterman algorithm, but also agree well with the known fact of evolution and results drawn by other authors [26, 27, 38–40]. In particular, compared with the conclusion of [27], the DV-curve representation reflecting the similarities of sequences is more simple, intuitional, and visible.The DV-curve graphical representations of different ND6 proteins.
(a)
(b)
(c)
(d)


### 4.2. The Phylogenetic Analysis among the Spike Glycoprotein of Coronaviruses

Coronaviruses belong to order Nidovirales, family Coronaviridae, and genusCoronavirus. They are a diverse group of large, enveloped, single-stranded RNA viruses that cause respiratory and enteric diseases in humans and other animals. Generally, coronaviruses can be divided into three groups: the first group and the second group come from mammalian; the third group comes from poultry (chicken and turkey). A novel coronavirus has been identified as the cause of the outbreak of severe acute respiratory syndrome (SARS). Previous phylogenetic analysis based on sequence alignments shows that SARS-CoVs come from a new group distantly related to the above three groups of previously characterized coronaviruses [41, 42]. The spike (S) protein, which is common to all known coronaviruses, is crucial for viral attachment and entry into the host cell. To illustrate the use of DV-curve of protein sequences, we will construct the phylogenetic tree of 35 coronavirus spike proteins of Table 1.Table 1
The information of 35 coronavirus spike proteins.
Number
Accession number
Abbreviation notation
Length (aa)
Group
1
P10033
FCoV1
1452
I
2
Q66928
FCoV2
1454
I
3
Q91AV1
PEDV3
1383
I
4
Q9DY22
TGEV4
1449
I
5
P18450
TGEV5
1449
I
6
P36300
CCoV6
1451
I
7
Q9J3E7
MHV7
1324
II
8
Q83331
MHV8
1361
II
9
P11224
MHV9
1324
II
10
O55253
MHV10
1360
II
11
Q9IKD1
RtCoV11
1360
II
12
P25190
BCoV12
1363
II
13
P15777
BCoV13
1363
II
14
Q9QAR5
BCoV14
1363
II
15
P36334
BCoV15
1363
II
16
P36334
HCoV16
1353
II
17
Q82666
IBV17
1166
III
18
P05135
IBV18
1163
III
19
P12722
IBV19
1154
III
20
Q64930
IBV20
1168
III
21
Q82624
IBV21
1159
III
22
P11223
IBV22
1162
III
23
Q98Y27
IBV23
1162
III
24
AAP41037
SCoV24
1255
IV
25
AAP300030
SCoV25
1255
IV
26
AAR91586
SCoV26
1255
IV
27
AAP51227
SCoV27
1255
IV
28
AAP33697
SCoV28
1255
IV
29
AAP13441
SCoV29
1255
IV
30
AAQ01597
SCoV30
1255
IV
31
AAU81608
SCoV31
1255
IV
32
AAS00003
SCoV32
1255
IV
33
AAR86788
SCoV33
1255
IV
34
AAR23250
SCoV34
1255
IV
35
AAT76147
SCoV35
1255
IVAs we have described above, a protein sequence can be associated with a 24-component vector. Given two speciesi and j, we can calculate the distance between them. Our datasets used in this paper were downloaded from GenBank (see Table 1 for details). Corresponding to 35 spike proteins, a 35
×
35 real symmetric matrix D
=
(
d
i
j
) is obtained and used to reflect the evolutionary distance of them. Using the UPGMA program included in PHYLIP package 3.65, we can construct the phylogenetic tree of these 35 species [43, 44]. The branch lengths are not scaled according to the distances and only the topology of the tree is concerned.Figure4 shows coronaviruses can be overall divided into four groups. Furthermore, it is evident that SARS-CoVs appear to cluster together and form a separate branch, which can be distinguished easily from the other three groups of coronaviruses.Figure 4
The phylogenetic tree based on the spike proteins.RtCoV11, MHV8, MHV10, HCoV16, BCoV13, BCoV12, BCoV15, BCoV14, MHV9, and MHV7, which belong to group 2, are situated at an independent branch, while TGEV5, FCoV2, CCoV6, TGEV4, FCoV1, and PEDV3, belonging to group 1, tend to cluster together. Meanwhile, the group 3 coronaviruses, including IBV22, IBV20, IBV23, IBV19, IBV18, IBV21, and IBV17, tend to cluster together in another branch. The resulting monophyletic clusters agree well with the established taxonomic groups [45, 46]. The conclusion is similar to that reported by other authors [23, 24]. Compared with result [24], it is noteworthy that a closer look at the subtree of the first branch shows coronavirus from three different species; that is, MHV, BCoV, and HCoV can be separated clearly, while they cluster together in a subtree by Li’s method. Obviously, our conclusion is more consistent with the known evolution fact.


## 4.1. Similarity Analysis Based on Visual Inspection of the Protein DV-Curve Graphs

Since Smith and Waterman developed a dynamic programming algorithm in 1981, many alignment algorithms identifying whether two biological sequences are similar to each other have been studied. These methods are proved to be efficient. However, multiple sequence alignment (MSA) of several hundred sequences has always produced a bottleneck.In 1994, MSA was proved to be an NP-complete problem by Wang and Jiang [36]. Moreover, most experts think that it is impossible until now to build a deterministic polynomial algorithm to handle an NP-complete problem. It needs to exhaust almost billions or trillions of years. Except long computational time, there also exists possible bias of multiple sequence alignments for multiple occurrences of highly similar sequence [37].However, our brain is much more powerful than computer when recognizing different figures. So it can help us to analyze the similarity in multiple sequences. If we can provide a simple, intuitional, clear, and nondegenerate 2D graphical representation of protein sequences, molecular biologists may easily find out which sequence is most similar or dissimilar to the given target sequence. And next they can use alignment algorithms for further confirmation.According to our proposed definition of protein DV-curve, we can draw the curves of some ND6 (NADH dehydrogenase subunit 6) proteins in order to conveniently compare them. Protein sequences that are used to prove our approach were downloaded from GenBank: human(
Y
P
_
003024037.1
), gorilla (
N
P
_
008223
), chimpanzee (
N
P
_
008197
), wallaroo (
N
P
_
007405
), harbor seal (H. seal) (
N
P
_
006939
), gray seal (G. seal) (
N
P
_
007080
), rat (
A
P
_
004903
), and mouse (
N
P
_
904339
), and the same data set was used in [26, 27].In Figure3, it is evident that protein graph of wallaroo is obviously different from the other species because it is the most remote species from the remaining mammals. Furthermore, we can see human and chimpanzee have similar curves, harbor seal and gray seal’s curves are almost identical, and two curves of rat and mouse are very similar. All these results not only are consistent with the conclusions drawn by Smith-Waterman algorithm, but also agree well with the known fact of evolution and results drawn by other authors [26, 27, 38–40]. In particular, compared with the conclusion of [27], the DV-curve representation reflecting the similarities of sequences is more simple, intuitional, and visible.The DV-curve graphical representations of different ND6 proteins.
(a)
(b)
(c)
(d)


## 4.2. The Phylogenetic Analysis among the Spike Glycoprotein of Coronaviruses

Coronaviruses belong to order Nidovirales, family Coronaviridae, and genusCoronavirus. They are a diverse group of large, enveloped, single-stranded RNA viruses that cause respiratory and enteric diseases in humans and other animals. Generally, coronaviruses can be divided into three groups: the first group and the second group come from mammalian; the third group comes from poultry (chicken and turkey). A novel coronavirus has been identified as the cause of the outbreak of severe acute respiratory syndrome (SARS). Previous phylogenetic analysis based on sequence alignments shows that SARS-CoVs come from a new group distantly related to the above three groups of previously characterized coronaviruses [41, 42]. The spike (S) protein, which is common to all known coronaviruses, is crucial for viral attachment and entry into the host cell. To illustrate the use of DV-curve of protein sequences, we will construct the phylogenetic tree of 35 coronavirus spike proteins of Table 1.Table 1
The information of 35 coronavirus spike proteins.
Number
Accession number
Abbreviation notation
Length (aa)
Group
1
P10033
FCoV1
1452
I
2
Q66928
FCoV2
1454
I
3
Q91AV1
PEDV3
1383
I
4
Q9DY22
TGEV4
1449
I
5
P18450
TGEV5
1449
I
6
P36300
CCoV6
1451
I
7
Q9J3E7
MHV7
1324
II
8
Q83331
MHV8
1361
II
9
P11224
MHV9
1324
II
10
O55253
MHV10
1360
II
11
Q9IKD1
RtCoV11
1360
II
12
P25190
BCoV12
1363
II
13
P15777
BCoV13
1363
II
14
Q9QAR5
BCoV14
1363
II
15
P36334
BCoV15
1363
II
16
P36334
HCoV16
1353
II
17
Q82666
IBV17
1166
III
18
P05135
IBV18
1163
III
19
P12722
IBV19
1154
III
20
Q64930
IBV20
1168
III
21
Q82624
IBV21
1159
III
22
P11223
IBV22
1162
III
23
Q98Y27
IBV23
1162
III
24
AAP41037
SCoV24
1255
IV
25
AAP300030
SCoV25
1255
IV
26
AAR91586
SCoV26
1255
IV
27
AAP51227
SCoV27
1255
IV
28
AAP33697
SCoV28
1255
IV
29
AAP13441
SCoV29
1255
IV
30
AAQ01597
SCoV30
1255
IV
31
AAU81608
SCoV31
1255
IV
32
AAS00003
SCoV32
1255
IV
33
AAR86788
SCoV33
1255
IV
34
AAR23250
SCoV34
1255
IV
35
AAT76147
SCoV35
1255
IVAs we have described above, a protein sequence can be associated with a 24-component vector. Given two speciesi and j, we can calculate the distance between them. Our datasets used in this paper were downloaded from GenBank (see Table 1 for details). Corresponding to 35 spike proteins, a 35
×
35 real symmetric matrix D
=
(
d
i
j
) is obtained and used to reflect the evolutionary distance of them. Using the UPGMA program included in PHYLIP package 3.65, we can construct the phylogenetic tree of these 35 species [43, 44]. The branch lengths are not scaled according to the distances and only the topology of the tree is concerned.Figure4 shows coronaviruses can be overall divided into four groups. Furthermore, it is evident that SARS-CoVs appear to cluster together and form a separate branch, which can be distinguished easily from the other three groups of coronaviruses.Figure 4
The phylogenetic tree based on the spike proteins.RtCoV11, MHV8, MHV10, HCoV16, BCoV13, BCoV12, BCoV15, BCoV14, MHV9, and MHV7, which belong to group 2, are situated at an independent branch, while TGEV5, FCoV2, CCoV6, TGEV4, FCoV1, and PEDV3, belonging to group 1, tend to cluster together. Meanwhile, the group 3 coronaviruses, including IBV22, IBV20, IBV23, IBV19, IBV18, IBV21, and IBV17, tend to cluster together in another branch. The resulting monophyletic clusters agree well with the established taxonomic groups [45, 46]. The conclusion is similar to that reported by other authors [23, 24]. Compared with result [24], it is noteworthy that a closer look at the subtree of the first branch shows coronavirus from three different species; that is, MHV, BCoV, and HCoV can be separated clearly, while they cluster together in a subtree by Li’s method. Obviously, our conclusion is more consistent with the known evolution fact.


## 5. Conclusion

According to the detailed hydrophobic-hydrophilic (HP) model of amino acids, we can reduce a protein primary sequence containing 20 amino acids into a four-letter sequence, which can be treated as a coarse-grained description of the protein primary sequence. Here we cannot avoid losing some information in the reduced sequences, but we can focus our main attention on the part of our interest.Some alignment-free methods to analyze DNA sequences have been proposed. However, there are few alignment-free methods to analyze protein sequences. Our method realizes the generalization from DNA graphical representations to those of proteins acceptable and can be seen a valid supplement to graphical representation of protein sequences. Meanwhile we first propose to combine DV-curve and the detailed HP model together to describe protein sequences.Compared with classical Smith-Waterman algorithm, the similarity/dissimilarity analysis results are consistent with DV-curve. In addition, the advantage of our method is that it can visualize the local and global features among different proteins no matter how long these sequences are and avoid degeneracy at the same time. The new approach is applied in two aspects: one is similarity intuitive analysis of ND6 protein sequences of several species and the other is phylogenetic analysis among 35 coronaviruses based on their spike proteins. Results have shown that our proposed method is more intuitional, simple, effectual, and feasible.



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*Source: 203871-2014-05-08.xml*