# ADLD: A Novel Graphical Representation of Protein Sequences and Its Application

Authors: Lei Wang; Hui Peng; Jinhua Zheng
Journal: Computational and Mathematical Methods in Medicine
 (2014)

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## Abstract

To facilitate the intuitional analysis of protein sequences, a novel graphical representation of protein sequences called ADLD (Alignment Diagonal Line Diagram) is introduced in this paper first, and then a new ADLD based method is proposed and utilized to analyze the similarity/dissimilarity of protein sequences. Comparing with existing methods, our ADLD based method is proved to be effective in the similarity/dissimilarity analysis of protein sequences and have the merits of good intuition, visuality, and simplicity. The examinations of the similarities/dissimilarities for both the 16 different ND5 proteins and the 29 different spike proteins illustrate the utility of our ADLD based approach.


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## Body

## 1. Introduction

Homology analysis is one of the hot topics in the area of protein sequences analysis. Up to now, lots of methods have been proposed for the homology analysis of protein sequences [1–3], and among them a useful one is the graphical representation of protein sequences, which is proved to be a powerful tool for visual comparison of protein sequences.At first, graphical representation methods were introduced for representation of DNA sequences on the basis of multiple dimension space [4–7]. After obtaining the sequence invariants from the graphics, one can compare the sequences based on comparison of sequence invariants. Graphical representation methods were proposed as an alternative approach of direct comparison of DNA sequences, which are computational intensive (even those of a restricted length) [8]. Protein sequences are to some degree similar to DNA sequences, which are composed of different units. Thus the graphical representation methods can be extended to describe protein sequences obviously.Currently, many researchers have proposed different methods for the graphical representation of protein sequences [9–24]. For example, Feng and Zhang [25] suggested Zp-curve based on the hydrophobicity and charged properties of amino acid residues along the primary sequence. Randić et al. [26] introduced a graphical representation of protein sequences based on a graphical representation of triplets of DNA in which the interior of a square or a tetrahedron is utilized to accommodate 64 sites for the 64 codons. Bai and Wang [27] derived a 2D graphical representation of protein sequences based on nucleotide triplet codons. Yao et al. [28] outlined a 2D graphical representation of protein sequences based on two classifications of amino acids. Abo el Maaty et al. [29] proposed a novel unique 3D graphical representation of protein sequences based on three physicochemical properties of amino acid side chains. Abo-Elkhier introduced a 3D graphical representation of protein sequence based on a right cone of a unit base and unit height on protein sequences interfaces [30]. El-Lakkani and El-Sherif [31] proposed a graphical representation of protein sequence to help similarity analysis of protein sequences based on 2D and 3D amino acid adjacency matrices. Ma et al. [32] introduced a family of Iterated Function Systems (IFS) to outline a 2D graphical representation of protein sequences.In most of these existing methods, the main drawbacks are that the higher the dimension of the protein sequence graphs, the heavier the computation complexity of the methods or the lower the recognition degree of the protein sequence graphs. For example, in the methods proposed in [26, 28], the main drawback is that the lines will cross each other, which will decrease the visibility of the graphics. In the methods proposed in [29–31], the main drawbacks are that the 3D graphics seem to be more complex and have lower visibility than the 2D graphics, and, in addition, to obtain the sequence invariants from the graphics, complex matrixes are required to be constructed, which need much computation and storage.Sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences [33]. Up to now, there are many kinds of algorithms having been implemented for sequence alignment [34–37]. These methods are usually efficient but complex and time consuming. Comparing with the alignment methods, existing graphical representation methods can also display the inner structure of the protein sequences and can be utilized to find the similarity/dissimilarity more visible according to their graphics. In this paper, we proposed a novel method for analyzing the similarity/dissimilarity by combining the idea of the sequence alignment and the graphical representation methods to some degree avoid the weakness of both of these two methods.Principal components analysis (PCA) is a standard tool in multivariate data analysis to reduce the number of dimensions, which has been proved to be effective in the process of protein sequence analysis [38–40]. Therefore, in order to overcome the main drawbacks of existing methods, in this paper, a novel graphical representation of protein sequences called ADLD (Alignment Diagonal Line Diagram) is introduced based on PCA, and then a new ADLD based method is proposed and utilized to analyze the similarity/dissimilarity of protein sequences. And, in addition, to validate the effectiveness of our ADLD based method, we adopt it to analyze the similarity/dissimilarity of both the 16 different ND5 proteins and the 29 different spike proteins, respectively, which are widely used as the test data [16–26]. The analysis results show that our method is not only visual, intuitional, and effective in the similarity/dissimilarity analysis of protein sequences but also quite simple, since there are no high dimensional matrixes required to be constructed.


## 2. Materials and Methods

### 2.1. Procedure of Our Method for Analysis of Protein Sequences

In this section, we will illustrate the overall procedures of our method for analyzing protein sequences as follows at first.(1)
Select the same 9 different properties for each amino acid and construct a 20 × 9 matrix as the input data of the PCA algorithm on the basis of total 20 different amino acids.(2)
According to the PCA algorithm, we can obtain a unique feature for each amino acid.(3)
For each protein sequence in the test data, we will replace each amino acid in the protein sequence with its corresponding unique feature, and then we can transform the protein sequence into a numerical sequence.(4)
For any two numerical sequences, we can draw a graph, named ADLD, and then abstract some numerical characteristics of it, which can be utilized to analyze the similarity/dissimilarity of these two sequences.Next, in Sections2.2–2.6 we will introduce the details of constructing the ADLDs and obtaining some of the numerical characteristics of them. In Section 3.1, we will give the method for constructing the similarity/dissimilarity of our test sequence groups.


### 2.2. Amino Acids and Their Properties

Proteins are composed of 20 different amino acids, and these amino acids have many different physicochemical and biological properties such as the molecular weight (mW), hydropathy index (hI), the pKa value for terminal amino acid groups COOH (pK1), the pKa value for terminal amino acid groups NH3+ (pK2), isoelectric point (pI), solubility (S), the number of triplet codons (cN), frequency of human proteins (F), and van der Waals radius of side chains (vR). The names and symbols of the 20 amino acids and the value of their 9 major properties are illustrated in Table 1.Table 1
The full list of 20 amino acids and the value of their 9 different properties.
Amino acid
Symbol
mW
hI
pK1
pK2
pI
S
cN
F (%)
vR
Alanine
A
89.079
1.8
2.34
9.69
6.01
167.2
4
7.8
67
Cysteine
C
121.145
2.5
1.96
10.28
5.07
0
2
1.9
86
Aspartic acid
D
133.089
−3.5
1.88
9.6
2.77
5
2
5.3
91
Glutamic acid
E
147.116
−3.5
2.19
9.67
3.22
8.5
2
6.3
109
Phenylalanine
F
165.177
2.8
1.83
9.13
5.48
27.6
2
3.9
135
Glycine
G
75.052
−0.4
2.34
9.6
5.97
249.9
4
7.2
48
Histidine
H
155.141
−3.2
1.82
9.17
7.59
0
2
2.3
118
Isoleucine
I
131.16
4.5
2.36
9.68
6.02
34.5
3
5.3
124
Lysine
K
146.17
−3.9
2.18
8.95
9.74
739
2
5.9
135
Leucine
L
131.16
3.8
2.36
9.6
5.98
21.7
6
9.1
124
Methionine
M
149.199
1.9
2.28
9.21
5.74
56.2
1
2.3
124
Asparagine
N
132.104
−3.5
2.02
8.8
5.41
28.5
2
4.3
96
Proline
P
115.117
1.6
1.99
10.96
6.48
1620
4
5.2
90
Glutamine
Q
146.131
−3.5
2.17
9.13
5.65
7.2
2
4.2
114
Arginine
R
174.188
−4.5
2.17
9.04
10.76
855.6
6
5.1
148
Serine
S
105.078
−0.8
2.21
9.15
5.68
422
6
6.8
73
Tyrosine
T
119.105
−0.7
2.11
9.62
5.87
13.2
4
5.9
93
Valine
V
117.133
4.2
2.32
9.62
5.97
58.1
4
6.6
105
Tryptophan
W
204.213
−0.9
2.38
9.39
5.89
13.6
1
1.4
163
Threonine
Y
181.176
−1.3
2.2
9.11
5.66
0.4
2
3.2
141


### 2.3. Principal Components Analysis

Principal components analysis (PCA) is a common technique for dimensionality reduction and pattern recognition in datasets of high dimension [41]. The main purposes of PCA are the analysis of data to identify patterns and finding patterns to reduce the dimensions of the dataset with minimal loss of information. The general steps of conducting PCA are as follows.Step 1.
Form samples {X1,X2,…,Xm}, suppose that each Xi has n components {xi1,xi1,…,xin}, let Xi=(xi1,xi1,…,xin) for i∈{1,2,…,m}, and then construct an m×n matrix X according to the following formula first:
(1)X=X1X2⋮Xm=x11x12⋯x1nx21x22⋯x2n⋮⋮⋮⋮xm1xm2⋯xmn.
Next, based on the matrixX, construct the corresponding m×n standardized matrix X* according to the following formula:
(2)X*=X1*X2*⋮Xm*=x11*x12*⋯x1n*x21*x22*⋯x2n*⋮⋮⋮⋮xm1*xm2*⋯xmn*,
where Xi*=(xi1*,xi2*,…,xin*), xij*=xij-x-j/var(xj), x-j=1/n∑i=1nxij, and var(xj)=1/n-1∑i=1n(xij-x-j)2, for i∈{1,2,…,m} and j∈{1,2,…,n}.Step 2.
Based on the matrixX*, construct the n×n correlation matrix R according to the following formula:
(3)R=R1R2⋮Rn=r11r12⋯r1nr21r22⋯r2n⋮⋮⋮⋮rn1rn2⋯rnn,
where we can find that rij=∑k=1m(xki*-x-i*)(xkj*-x-j*)/∑k=1m(xki*-x-i*)2∑k=1m(xkj*-x-j*)2 for i∈{1,2,…,n} and j∈{1,2,…,n}.Step 3.
From the correlation matrixR, obtain its n eigenvalues λ1≥λ2≥⋯≥λn>0 and the corresponding n eigenvectors
(4)a1=a11a21⋮an1,a2=a12a22⋮an2,…,an=a1na2n⋮ann,
respectively. And, from now on, we can obtain n principal components Fi for i∈{1,2,…,n} as follows:
(5)Fi=a1iX1*+a2iX2*+⋯+aniXn*.Step 4.
For each principal componentFi for i∈{1,2,…,n}, obtain itscontribution rateCRi andaccumulated contribution rateACRi according to the following formulas, respectively:
(6)CRi=λi∑k=1nλk,(7)ACRi=∑k=1iCRk.
Generally, in order to lower the computation complexity, we can keep only the firstt (t≤n) principal components {F1,F2,…,Ft}, where the accumulated contribution rate of the tth principal component Ft shall satisfy the fact that ACRt≥85%.Step 5.
Forj∈{1,2,…,t}, let
(8)Fj=F1jF2j⋮Fmj.
Then, for each i∈{1,2,…,m}, we can obtain the total score of the ith sample as follows:
(9)TotalScorei=∑k=1tFik×CRk.


### 2.4. PCA of the Amino Acids

Observing Table1, if we consider the 20 amino acids as 20 different samples and the 9 properties of each amino acid as its 9 components, then, according to the general steps of conducting PCA illustrated in Section 2.3, we can obtain a 20 × 9 matrix X and its standardized matrix X*, a 9 × 9 correlation matrix R, and 9 principal components {F1,F2,…,F9}. And, therefore, as illustrated in Table 2, we can obtain the 9 eigenvalues of R and the contribution rates and the accumulative contribution rates of the 9 principal components {F1,F2,…,F9}, respectively.Table 2
The 9 eigenvalues (λ) of R and the contribution rates (CR) and the accumulative contribution rates (ACR) of the 9 principal components obtained by conducting PCA of the 20 amino acids.
Number
λ
CR
ACR
1
3.2237
0.3582
0.3582
2
1.9132
0.2126
0.5708
3
1.4048
0.1561
0.7269
4
1.1876
0.1320
0.8588
5
0.4959
0.0551
0.9139
6
0.4467
0.0496
0.9635
7
0.1992
0.0221
0.9857
8
0.1218
0.0135
0.9992
9
0.0071
0.0008
1.0000From Table2, we can see that the accumulative contribution rate of the first 4 principal components amounts to 0.8588 (=85.88%), which is already bigger than 85%. Therefore, we can keep the first 4 principal components only. Let {λ1,λ2,λ3,λ4} be the 4 eigenvalues corresponding to the first 4 principal components, respectively; then, as illustrated in Table 3, we can obtain the 4 eigenvectors {a1,a2,a3,a4} corresponding to the 4 eigenvalues {λ1,λ2,λ3,λ4} separately.Table 3
The 4 eigenvectors{a1,a2,a3,a4} corresponding to the first 4 eigenvalues in Table 2.
a
1
a
2
a
3
a
4
0.5036
0.1436
0.0571
0.2158
−0.2454
−0.1875
0.2304
0.6547
−0.1634
0.1820
0.6298
0.2288
−0.3101
−0.1883
−0.3964
0.5071
0.0702
0.6464
−0.0786
0.0532
−0.1665
0.4465
−0.5280
0.1877
−0.3872
0.3931
0.1003
−0.0532
−0.4377
0.1844
0.2544
−0.2273
0.4349
0.2643
0.1738
0.3495Based on Table3, we can obtain the first 4 principal components {F1,F2,F3,F4} as follows:
(10)F1=0.5036X1*-0.2454X2*-0.1634X3*-0.3101X4*+0.0702X5*-0.1665X6*-0.3872X7*-0.4377X8*+0.4349X9*,F2=0.1436X1*-0.1875X2*+0.1820X3*-0.1883X4*+0.6464X5*+0.4465X6*+0.3931X7*+0.1844X8*+0.2643X9*,F3=0.0571X1*+0.2304X2*+0.6298X3*-0.3964X4*-0.0786X5*-0.5280X6*+0.1003X7*+0.2544X8*+0.1738X9*,F4=0.2158X1*+0.6547X2*+0.2288X3*+0.5071X4*+0.0532X5*+0.1877X6*-0.0532X7*-0.2273X8*+0.3495X9*.Observing the above 4 formulas, it is easy to find that there are three big coefficients in the first formula, which are 0.5036 (corresponding tomW), 0.4377 (corresponding to F), and 0.4349 (corresponding to vR), respectively. Therefore, it means that the three properties such as mW, F, and vR will have a major role in the first principal component F1. Similarly, we can also know that the three properties such as pI, S, and cN will have a major role in the second principal component F2, the third principal component F3 is mainly determined by pK1 and S, and the fourth principal component F4 is closely linked with hI and pK2 and so forth. Hence, we can obtain the total scores of the 20 amino acids as illustrated in Table 4 according to formula (9).Table 4
The total scores of the 20 amino acids.
Symbols of amino acids
Total scores
A
−0.9324
C
−0.5985
D
−0.6709
E
−0.2296
F
0.4298
G
−1.1780
H
0.4476
I
0.1435
K
0.7868
L
−0.1205
M
0.5735
N
−0.0242
P
−0.9822
Q
0.2848
R
1.1169
S
−0.7077
T
−0.4525
V
−0.2643
W
1.4729
Y
0.9050


### 2.5. Numerical Sequences of Protein Sequences

LetΩ={A,C,D,E,F,G,H,I,K,L,M,N,P,Q,R,S,T,V,W,Y} and suppose that Ψ=p1p2p3…pN represents a protein sequence with N amino acids, where pi∈Ω for i∈{1,2,…,N}; then we can obtain a numerical sequence SΨ=(t1,t2,…,tN) corresponding to the protein sequence Ψ through replacing each amino acid pi in Ψ with its corresponding value of TotalScore(i) for i∈{1,2,…,N}.For example, consider the following 3 abbreviated protein sequences:Hu = MTMHTTMTTL,Gor = MTMYATMTTL,Opo = MKVINISNTM.According to the above descriptions and Table4, then we can obtain their corresponding numerical sequences as follows:
(11)SHu=0.5735,-0.4525,0.5735,0.4476,-0.4525,-0.4525,iii0.5735,-0.4525,-0.4525,-0.1205;SGor=0.5735,-0.4525,0.5735,0.9050,-0.9324,-0.4525,iii0.5735,-0.4525,-0.4525,-0.1205;SOpo=0.5735,0.7868,-0.2643,0.1435,-0.0242,0.1435,iii-0.7077,-0.0242,-0.4525,0.5735.


### 2.6. ASDs and ADLDs of Protein Sequence Pairs

For a given protein sequence pair (s1,s2), suppose that the protein sequence s1 includes N1 amino acids, s2 includes N2 amino acids, and N1⩾N2; then, in order to measure the similarity/dissimilarity between them, in this section, we will present a new method calledAlignment Scatter Diagram (ASD) to plot the two sequences into a scatter diagram first. And, for convenience, we call the points in the ASD thealignment-plots (APs). The ASD of the protein sequence pair (s1,s2) can be obtained through the following steps.Step 1.
According to the method given in Section2.5, translate the protein sequence pair (s1,s2) into two numerical sequences with the same length as follows:
(12)S1=t1,t2,…,tN1,S2=t1,t2,…,tN2,0,0,…,0︸N1-N2.Step 2.
Letw be thealignment width (AW) of the protein sequence pair (s1,s2); that is, let s1=p1,p2,p3,…,pN1, s2=q1,q2,q3,…,qN2; then, for any amino acid pi in the protein sequence s1, we will compare it with these 2w+1 amino acids {qi-w,…,qi-1,qi,qi+1,…,qi+w} in the protein sequence s2, and then w can be simply defined as follows:
(13)w=ξ,ifN1-N2≤ξ,N1-N2,else,
where ξ>0 is a given threshold to guarantee that the AW of the protein sequence pair (s1,s2) will not be too small to expose the association of the inner structures of the protein sequence pair (s1,s2). In actual applications, we suggest that ξ shall be no less than 10.Step 3.
Letε>0 be thedissimilarity degree (DD) of two amino acids; that is, if ε=0, then it means that the two amino acids are the same; otherwise, it means that the two amino acids are different from each other to some degree, and then the APs in the ASD of the protein sequence pair (s1,s2) can be briefly defined as follows:
(14)Aijε=Θti-tj-ε,
where i∈{1,2,…,N1}, j∈{1,2,…,N1}, and Θ is a Heaviside function, which can be defined as follows:
(15)Θx=1,ifx≤0,0,else.
Thereafter, we can obtain anN1×N1alignment matrix (AM) as follows:
(16)AM=AijεN1×N1.Step 4.
For theN1×N1 elements in the alignment matrix AM, we can plot points on i-j plane for these elements in the AM with Aijε=1 and |i-j|≤w. And, for convenience, we call the obtained graph theAlignment Scatter Diagram (ASD) of the protein sequence pair (s1,s2).For example, considering the threeβ-globin protein sequences of chimpanzee [GenBank: AAA16334.1], human [GenBank: CAA26204.1], and gorilla [GenBank: CAA43421.1] obtained from the GenBank, respectively, we illustrate the ASDs of the β-globin protein sequence pair (chimpanzee, human) and the β-globin protein sequence pair (human, gorilla) in Figures 1(a) and 1(b) separately while letting ε=0.(a) The ASD of theβ-globin protein sequence pair (chimpanzee, human) with ξ=12; (b) the ASD of the β-globin protein sequence pair (human, gorilla) with ξ=16.
(a)
(b)From Figure1, it is easy to see that there are lots of disordered points in these ASDs, which will lower the visuality of the ASDs remarkably and obstruct us from distinguishing the similarity/dissimilarity between the protein sequence pairs intuitively while observing these ASDs. Therefore, in order to improve the intuition of the ASD, we will propose a simplified variant diagram of the ASD, which is called theAlignment Diagonal Line Diagram (ADLD).For convenience, in an ASD, we call its main diagonal line theartery tracks (ATs) and the lines parallelling to its main diagonal line theby-path tracks (BTs), respectively. And, in addition, we define a set consisting with no less than δ consecutive APs on the AT or BTs as a CAPS, where δ≥1 is a given threshold.For a given CAPS caps1, if there is no CAPS caps2 satisfying caps1⊂caps2, then we call the caps1 a maximum CAPS. And, for convenience, we call the line formed by connecting all of the APs in a maximum CAPS asimilar fragment (SF), and simultaneously we call all of the APs on the AT but not on any SFs thefree points (FPs).Obviously, in an ASD, if keeping all of the SFs and FPs only and omitting all those other APs, then we will obtain a simplified variant diagram of the ASD, and, for convenience, we call it theAlignment Diagonal Line Diagram (ADLD). Apparently, if δ=1, then an ADLD will degenerate into an ASD. Therefore, in actual applications, we suggest that δ will be no less than 2. And, particularly, in order to find more accurate SFs in the ADLD of a protein sequence pair, the longer the protein sequences in the protein sequence pair are the bigger the value of δ shall be.For convenience of analysis, in an ADLD, suppose that there areK1 different SFs and K2 different FPs on its AT, K different BTs locating above its AT, and K different BTs locating below its AT; then we get the following.(1)
For theseK1 different SFs and K2 different FPs on the AT of the ADLD, we will number these K1 SFs and K2 FPs from left to right and utilize {ASF1,ASF2,…,ASFK1} and {FP1,FP2,…,FPK2} to represent these K1 SFs and K2 FPs separately. And, in addition, we would also call these SFs on the AT of the ADLD the ASFs.(2)
For theseK different BTs locating above the AT, we will number these BTs from down to up and utilize {BT1,BT2,…,BTK} to represent these BTs separately, and, for these K different BTs locating below the AT, we will number these BTs from up to down and utilize {BT-1,BT-2,…,BT-K} to represent these BTs separately.(3)
For eachBTl, where l∈{1,2,…,K}, suppose that there are K3 different SFs on the BTl; then we will number these K3 SFs from left to right and utilize {BSFl1,BSFl2,…,BSFlK3} to represent these SFs separately. And, in addition, we would also call these SFs on the BTs of the ADLD the BSFs.According to the above assumptions, in Figure2, we show the two ADLDs corresponding to the ASDs illustrated in Figures 1(a) and 1(b) while letting δ=3. And, in addition, to make the ADLDs more visual and intuitional, in Figure 2, we use the red “*” to represent the FPs on the AT and the blue lines to represent the SFs on the AT or BTs.(a) The ADLD of the protein sequence pair (chimpanzee, human); (b) the ADLD of the protein sequence pair (human, gorilla).
(a)
(b)From Figure2(a), it is easy to see that there are two SFs in the ADLD of the sequence pair (chimpanzee, human); one is ASF1, that is, the line segment from the point (1,1) to the point (32,32), and the other is BSF-41, that is, the line segment from the point (35,31) to the point (125,121). And, in addition, there are totally 6 FPs in the ADLD, which are FP1(46,46), FP2(66,66), FP3(111,111), FP4(114,114), FP5(115,115), and FP6(123,123), respectively.Observing Figure2(b), we can easily find that there are also two SFs in the ADLD of the sequence pair (human, gorilla). But, different from that in Figure 2(a), the two SFs in Figure 2(b) are both ASFs; one is ASF1, that is, the line segment from the point (1,1) to the point (104,104), and the other is ASF2, that is, the line segment from the point (106,106) to the point (121,121). And, in addition, the two ASFs in Figure 2(b) are separated by one gap, and there exist no FPs or BSFs on the AT or BTs.Through analysis, we can know that, for a given protein sequence pair, if there exist some deletions or insertions of amino acid segments between the two protein sequences, then there will exist some misalignments of SFs in their ADLD; that is, some ASFs on the AT will be transformed into BSFs on some BTs. And, in addition, if there exist some substitutions of the amino acids between the two protein sequences, then, in their ADLD, there will exist some gaps between two neighboring SFs or FPs on the AT. Furthermore, if there exist some insertions, deletions, or substitutions of the amino acid segments at the end of the two protein sequences, then, in their ADLD, there will exist no SFs or FPs on the AT or BTs.From the above descriptions, it is easy to know that the ADLD of any given protein sequence pair obtained by our above proposed method reflects some inner and specific differences between these two protein sequences in the given protein sequence pair, which may be useful in the similarity/dissimilarity analysis of protein sequence pairs.


## 2.1. Procedure of Our Method for Analysis of Protein Sequences

In this section, we will illustrate the overall procedures of our method for analyzing protein sequences as follows at first.(1)
Select the same 9 different properties for each amino acid and construct a 20 × 9 matrix as the input data of the PCA algorithm on the basis of total 20 different amino acids.(2)
According to the PCA algorithm, we can obtain a unique feature for each amino acid.(3)
For each protein sequence in the test data, we will replace each amino acid in the protein sequence with its corresponding unique feature, and then we can transform the protein sequence into a numerical sequence.(4)
For any two numerical sequences, we can draw a graph, named ADLD, and then abstract some numerical characteristics of it, which can be utilized to analyze the similarity/dissimilarity of these two sequences.Next, in Sections2.2–2.6 we will introduce the details of constructing the ADLDs and obtaining some of the numerical characteristics of them. In Section 3.1, we will give the method for constructing the similarity/dissimilarity of our test sequence groups.


## 2.2. Amino Acids and Their Properties

Proteins are composed of 20 different amino acids, and these amino acids have many different physicochemical and biological properties such as the molecular weight (mW), hydropathy index (hI), the pKa value for terminal amino acid groups COOH (pK1), the pKa value for terminal amino acid groups NH3+ (pK2), isoelectric point (pI), solubility (S), the number of triplet codons (cN), frequency of human proteins (F), and van der Waals radius of side chains (vR). The names and symbols of the 20 amino acids and the value of their 9 major properties are illustrated in Table 1.Table 1
The full list of 20 amino acids and the value of their 9 different properties.
Amino acid
Symbol
mW
hI
pK1
pK2
pI
S
cN
F (%)
vR
Alanine
A
89.079
1.8
2.34
9.69
6.01
167.2
4
7.8
67
Cysteine
C
121.145
2.5
1.96
10.28
5.07
0
2
1.9
86
Aspartic acid
D
133.089
−3.5
1.88
9.6
2.77
5
2
5.3
91
Glutamic acid
E
147.116
−3.5
2.19
9.67
3.22
8.5
2
6.3
109
Phenylalanine
F
165.177
2.8
1.83
9.13
5.48
27.6
2
3.9
135
Glycine
G
75.052
−0.4
2.34
9.6
5.97
249.9
4
7.2
48
Histidine
H
155.141
−3.2
1.82
9.17
7.59
0
2
2.3
118
Isoleucine
I
131.16
4.5
2.36
9.68
6.02
34.5
3
5.3
124
Lysine
K
146.17
−3.9
2.18
8.95
9.74
739
2
5.9
135
Leucine
L
131.16
3.8
2.36
9.6
5.98
21.7
6
9.1
124
Methionine
M
149.199
1.9
2.28
9.21
5.74
56.2
1
2.3
124
Asparagine
N
132.104
−3.5
2.02
8.8
5.41
28.5
2
4.3
96
Proline
P
115.117
1.6
1.99
10.96
6.48
1620
4
5.2
90
Glutamine
Q
146.131
−3.5
2.17
9.13
5.65
7.2
2
4.2
114
Arginine
R
174.188
−4.5
2.17
9.04
10.76
855.6
6
5.1
148
Serine
S
105.078
−0.8
2.21
9.15
5.68
422
6
6.8
73
Tyrosine
T
119.105
−0.7
2.11
9.62
5.87
13.2
4
5.9
93
Valine
V
117.133
4.2
2.32
9.62
5.97
58.1
4
6.6
105
Tryptophan
W
204.213
−0.9
2.38
9.39
5.89
13.6
1
1.4
163
Threonine
Y
181.176
−1.3
2.2
9.11
5.66
0.4
2
3.2
141


## 2.3. Principal Components Analysis

Principal components analysis (PCA) is a common technique for dimensionality reduction and pattern recognition in datasets of high dimension [41]. The main purposes of PCA are the analysis of data to identify patterns and finding patterns to reduce the dimensions of the dataset with minimal loss of information. The general steps of conducting PCA are as follows.Step 1.
Form samples {X1,X2,…,Xm}, suppose that each Xi has n components {xi1,xi1,…,xin}, let Xi=(xi1,xi1,…,xin) for i∈{1,2,…,m}, and then construct an m×n matrix X according to the following formula first:
(1)X=X1X2⋮Xm=x11x12⋯x1nx21x22⋯x2n⋮⋮⋮⋮xm1xm2⋯xmn.
Next, based on the matrixX, construct the corresponding m×n standardized matrix X* according to the following formula:
(2)X*=X1*X2*⋮Xm*=x11*x12*⋯x1n*x21*x22*⋯x2n*⋮⋮⋮⋮xm1*xm2*⋯xmn*,
where Xi*=(xi1*,xi2*,…,xin*), xij*=xij-x-j/var(xj), x-j=1/n∑i=1nxij, and var(xj)=1/n-1∑i=1n(xij-x-j)2, for i∈{1,2,…,m} and j∈{1,2,…,n}.Step 2.
Based on the matrixX*, construct the n×n correlation matrix R according to the following formula:
(3)R=R1R2⋮Rn=r11r12⋯r1nr21r22⋯r2n⋮⋮⋮⋮rn1rn2⋯rnn,
where we can find that rij=∑k=1m(xki*-x-i*)(xkj*-x-j*)/∑k=1m(xki*-x-i*)2∑k=1m(xkj*-x-j*)2 for i∈{1,2,…,n} and j∈{1,2,…,n}.Step 3.
From the correlation matrixR, obtain its n eigenvalues λ1≥λ2≥⋯≥λn>0 and the corresponding n eigenvectors
(4)a1=a11a21⋮an1,a2=a12a22⋮an2,…,an=a1na2n⋮ann,
respectively. And, from now on, we can obtain n principal components Fi for i∈{1,2,…,n} as follows:
(5)Fi=a1iX1*+a2iX2*+⋯+aniXn*.Step 4.
For each principal componentFi for i∈{1,2,…,n}, obtain itscontribution rateCRi andaccumulated contribution rateACRi according to the following formulas, respectively:
(6)CRi=λi∑k=1nλk,(7)ACRi=∑k=1iCRk.
Generally, in order to lower the computation complexity, we can keep only the firstt (t≤n) principal components {F1,F2,…,Ft}, where the accumulated contribution rate of the tth principal component Ft shall satisfy the fact that ACRt≥85%.Step 5.
Forj∈{1,2,…,t}, let
(8)Fj=F1jF2j⋮Fmj.
Then, for each i∈{1,2,…,m}, we can obtain the total score of the ith sample as follows:
(9)TotalScorei=∑k=1tFik×CRk.


## 2.4. PCA of the Amino Acids

Observing Table1, if we consider the 20 amino acids as 20 different samples and the 9 properties of each amino acid as its 9 components, then, according to the general steps of conducting PCA illustrated in Section 2.3, we can obtain a 20 × 9 matrix X and its standardized matrix X*, a 9 × 9 correlation matrix R, and 9 principal components {F1,F2,…,F9}. And, therefore, as illustrated in Table 2, we can obtain the 9 eigenvalues of R and the contribution rates and the accumulative contribution rates of the 9 principal components {F1,F2,…,F9}, respectively.Table 2
The 9 eigenvalues (λ) of R and the contribution rates (CR) and the accumulative contribution rates (ACR) of the 9 principal components obtained by conducting PCA of the 20 amino acids.
Number
λ
CR
ACR
1
3.2237
0.3582
0.3582
2
1.9132
0.2126
0.5708
3
1.4048
0.1561
0.7269
4
1.1876
0.1320
0.8588
5
0.4959
0.0551
0.9139
6
0.4467
0.0496
0.9635
7
0.1992
0.0221
0.9857
8
0.1218
0.0135
0.9992
9
0.0071
0.0008
1.0000From Table2, we can see that the accumulative contribution rate of the first 4 principal components amounts to 0.8588 (=85.88%), which is already bigger than 85%. Therefore, we can keep the first 4 principal components only. Let {λ1,λ2,λ3,λ4} be the 4 eigenvalues corresponding to the first 4 principal components, respectively; then, as illustrated in Table 3, we can obtain the 4 eigenvectors {a1,a2,a3,a4} corresponding to the 4 eigenvalues {λ1,λ2,λ3,λ4} separately.Table 3
The 4 eigenvectors{a1,a2,a3,a4} corresponding to the first 4 eigenvalues in Table 2.
a
1
a
2
a
3
a
4
0.5036
0.1436
0.0571
0.2158
−0.2454
−0.1875
0.2304
0.6547
−0.1634
0.1820
0.6298
0.2288
−0.3101
−0.1883
−0.3964
0.5071
0.0702
0.6464
−0.0786
0.0532
−0.1665
0.4465
−0.5280
0.1877
−0.3872
0.3931
0.1003
−0.0532
−0.4377
0.1844
0.2544
−0.2273
0.4349
0.2643
0.1738
0.3495Based on Table3, we can obtain the first 4 principal components {F1,F2,F3,F4} as follows:
(10)F1=0.5036X1*-0.2454X2*-0.1634X3*-0.3101X4*+0.0702X5*-0.1665X6*-0.3872X7*-0.4377X8*+0.4349X9*,F2=0.1436X1*-0.1875X2*+0.1820X3*-0.1883X4*+0.6464X5*+0.4465X6*+0.3931X7*+0.1844X8*+0.2643X9*,F3=0.0571X1*+0.2304X2*+0.6298X3*-0.3964X4*-0.0786X5*-0.5280X6*+0.1003X7*+0.2544X8*+0.1738X9*,F4=0.2158X1*+0.6547X2*+0.2288X3*+0.5071X4*+0.0532X5*+0.1877X6*-0.0532X7*-0.2273X8*+0.3495X9*.Observing the above 4 formulas, it is easy to find that there are three big coefficients in the first formula, which are 0.5036 (corresponding tomW), 0.4377 (corresponding to F), and 0.4349 (corresponding to vR), respectively. Therefore, it means that the three properties such as mW, F, and vR will have a major role in the first principal component F1. Similarly, we can also know that the three properties such as pI, S, and cN will have a major role in the second principal component F2, the third principal component F3 is mainly determined by pK1 and S, and the fourth principal component F4 is closely linked with hI and pK2 and so forth. Hence, we can obtain the total scores of the 20 amino acids as illustrated in Table 4 according to formula (9).Table 4
The total scores of the 20 amino acids.
Symbols of amino acids
Total scores
A
−0.9324
C
−0.5985
D
−0.6709
E
−0.2296
F
0.4298
G
−1.1780
H
0.4476
I
0.1435
K
0.7868
L
−0.1205
M
0.5735
N
−0.0242
P
−0.9822
Q
0.2848
R
1.1169
S
−0.7077
T
−0.4525
V
−0.2643
W
1.4729
Y
0.9050


## 2.5. Numerical Sequences of Protein Sequences

LetΩ={A,C,D,E,F,G,H,I,K,L,M,N,P,Q,R,S,T,V,W,Y} and suppose that Ψ=p1p2p3…pN represents a protein sequence with N amino acids, where pi∈Ω for i∈{1,2,…,N}; then we can obtain a numerical sequence SΨ=(t1,t2,…,tN) corresponding to the protein sequence Ψ through replacing each amino acid pi in Ψ with its corresponding value of TotalScore(i) for i∈{1,2,…,N}.For example, consider the following 3 abbreviated protein sequences:Hu = MTMHTTMTTL,Gor = MTMYATMTTL,Opo = MKVINISNTM.According to the above descriptions and Table4, then we can obtain their corresponding numerical sequences as follows:
(11)SHu=0.5735,-0.4525,0.5735,0.4476,-0.4525,-0.4525,iii0.5735,-0.4525,-0.4525,-0.1205;SGor=0.5735,-0.4525,0.5735,0.9050,-0.9324,-0.4525,iii0.5735,-0.4525,-0.4525,-0.1205;SOpo=0.5735,0.7868,-0.2643,0.1435,-0.0242,0.1435,iii-0.7077,-0.0242,-0.4525,0.5735.


## 2.6. ASDs and ADLDs of Protein Sequence Pairs

For a given protein sequence pair (s1,s2), suppose that the protein sequence s1 includes N1 amino acids, s2 includes N2 amino acids, and N1⩾N2; then, in order to measure the similarity/dissimilarity between them, in this section, we will present a new method calledAlignment Scatter Diagram (ASD) to plot the two sequences into a scatter diagram first. And, for convenience, we call the points in the ASD thealignment-plots (APs). The ASD of the protein sequence pair (s1,s2) can be obtained through the following steps.Step 1.
According to the method given in Section2.5, translate the protein sequence pair (s1,s2) into two numerical sequences with the same length as follows:
(12)S1=t1,t2,…,tN1,S2=t1,t2,…,tN2,0,0,…,0︸N1-N2.Step 2.
Letw be thealignment width (AW) of the protein sequence pair (s1,s2); that is, let s1=p1,p2,p3,…,pN1, s2=q1,q2,q3,…,qN2; then, for any amino acid pi in the protein sequence s1, we will compare it with these 2w+1 amino acids {qi-w,…,qi-1,qi,qi+1,…,qi+w} in the protein sequence s2, and then w can be simply defined as follows:
(13)w=ξ,ifN1-N2≤ξ,N1-N2,else,
where ξ>0 is a given threshold to guarantee that the AW of the protein sequence pair (s1,s2) will not be too small to expose the association of the inner structures of the protein sequence pair (s1,s2). In actual applications, we suggest that ξ shall be no less than 10.Step 3.
Letε>0 be thedissimilarity degree (DD) of two amino acids; that is, if ε=0, then it means that the two amino acids are the same; otherwise, it means that the two amino acids are different from each other to some degree, and then the APs in the ASD of the protein sequence pair (s1,s2) can be briefly defined as follows:
(14)Aijε=Θti-tj-ε,
where i∈{1,2,…,N1}, j∈{1,2,…,N1}, and Θ is a Heaviside function, which can be defined as follows:
(15)Θx=1,ifx≤0,0,else.
Thereafter, we can obtain anN1×N1alignment matrix (AM) as follows:
(16)AM=AijεN1×N1.Step 4.
For theN1×N1 elements in the alignment matrix AM, we can plot points on i-j plane for these elements in the AM with Aijε=1 and |i-j|≤w. And, for convenience, we call the obtained graph theAlignment Scatter Diagram (ASD) of the protein sequence pair (s1,s2).For example, considering the threeβ-globin protein sequences of chimpanzee [GenBank: AAA16334.1], human [GenBank: CAA26204.1], and gorilla [GenBank: CAA43421.1] obtained from the GenBank, respectively, we illustrate the ASDs of the β-globin protein sequence pair (chimpanzee, human) and the β-globin protein sequence pair (human, gorilla) in Figures 1(a) and 1(b) separately while letting ε=0.(a) The ASD of theβ-globin protein sequence pair (chimpanzee, human) with ξ=12; (b) the ASD of the β-globin protein sequence pair (human, gorilla) with ξ=16.
(a)
(b)From Figure1, it is easy to see that there are lots of disordered points in these ASDs, which will lower the visuality of the ASDs remarkably and obstruct us from distinguishing the similarity/dissimilarity between the protein sequence pairs intuitively while observing these ASDs. Therefore, in order to improve the intuition of the ASD, we will propose a simplified variant diagram of the ASD, which is called theAlignment Diagonal Line Diagram (ADLD).For convenience, in an ASD, we call its main diagonal line theartery tracks (ATs) and the lines parallelling to its main diagonal line theby-path tracks (BTs), respectively. And, in addition, we define a set consisting with no less than δ consecutive APs on the AT or BTs as a CAPS, where δ≥1 is a given threshold.For a given CAPS caps1, if there is no CAPS caps2 satisfying caps1⊂caps2, then we call the caps1 a maximum CAPS. And, for convenience, we call the line formed by connecting all of the APs in a maximum CAPS asimilar fragment (SF), and simultaneously we call all of the APs on the AT but not on any SFs thefree points (FPs).Obviously, in an ASD, if keeping all of the SFs and FPs only and omitting all those other APs, then we will obtain a simplified variant diagram of the ASD, and, for convenience, we call it theAlignment Diagonal Line Diagram (ADLD). Apparently, if δ=1, then an ADLD will degenerate into an ASD. Therefore, in actual applications, we suggest that δ will be no less than 2. And, particularly, in order to find more accurate SFs in the ADLD of a protein sequence pair, the longer the protein sequences in the protein sequence pair are the bigger the value of δ shall be.For convenience of analysis, in an ADLD, suppose that there areK1 different SFs and K2 different FPs on its AT, K different BTs locating above its AT, and K different BTs locating below its AT; then we get the following.(1)
For theseK1 different SFs and K2 different FPs on the AT of the ADLD, we will number these K1 SFs and K2 FPs from left to right and utilize {ASF1,ASF2,…,ASFK1} and {FP1,FP2,…,FPK2} to represent these K1 SFs and K2 FPs separately. And, in addition, we would also call these SFs on the AT of the ADLD the ASFs.(2)
For theseK different BTs locating above the AT, we will number these BTs from down to up and utilize {BT1,BT2,…,BTK} to represent these BTs separately, and, for these K different BTs locating below the AT, we will number these BTs from up to down and utilize {BT-1,BT-2,…,BT-K} to represent these BTs separately.(3)
For eachBTl, where l∈{1,2,…,K}, suppose that there are K3 different SFs on the BTl; then we will number these K3 SFs from left to right and utilize {BSFl1,BSFl2,…,BSFlK3} to represent these SFs separately. And, in addition, we would also call these SFs on the BTs of the ADLD the BSFs.According to the above assumptions, in Figure2, we show the two ADLDs corresponding to the ASDs illustrated in Figures 1(a) and 1(b) while letting δ=3. And, in addition, to make the ADLDs more visual and intuitional, in Figure 2, we use the red “*” to represent the FPs on the AT and the blue lines to represent the SFs on the AT or BTs.(a) The ADLD of the protein sequence pair (chimpanzee, human); (b) the ADLD of the protein sequence pair (human, gorilla).
(a)
(b)From Figure2(a), it is easy to see that there are two SFs in the ADLD of the sequence pair (chimpanzee, human); one is ASF1, that is, the line segment from the point (1,1) to the point (32,32), and the other is BSF-41, that is, the line segment from the point (35,31) to the point (125,121). And, in addition, there are totally 6 FPs in the ADLD, which are FP1(46,46), FP2(66,66), FP3(111,111), FP4(114,114), FP5(115,115), and FP6(123,123), respectively.Observing Figure2(b), we can easily find that there are also two SFs in the ADLD of the sequence pair (human, gorilla). But, different from that in Figure 2(a), the two SFs in Figure 2(b) are both ASFs; one is ASF1, that is, the line segment from the point (1,1) to the point (104,104), and the other is ASF2, that is, the line segment from the point (106,106) to the point (121,121). And, in addition, the two ASFs in Figure 2(b) are separated by one gap, and there exist no FPs or BSFs on the AT or BTs.Through analysis, we can know that, for a given protein sequence pair, if there exist some deletions or insertions of amino acid segments between the two protein sequences, then there will exist some misalignments of SFs in their ADLD; that is, some ASFs on the AT will be transformed into BSFs on some BTs. And, in addition, if there exist some substitutions of the amino acids between the two protein sequences, then, in their ADLD, there will exist some gaps between two neighboring SFs or FPs on the AT. Furthermore, if there exist some insertions, deletions, or substitutions of the amino acid segments at the end of the two protein sequences, then, in their ADLD, there will exist no SFs or FPs on the AT or BTs.From the above descriptions, it is easy to know that the ADLD of any given protein sequence pair obtained by our above proposed method reflects some inner and specific differences between these two protein sequences in the given protein sequence pair, which may be useful in the similarity/dissimilarity analysis of protein sequence pairs.


## 3. Results and Discussion

### 3.1. Method for Similarity/Dissimilarity Analysis of Protein Sequences Based on the ADLDs

According to the above analysis, we have known that the ADLDs may be useful in analyzing the differences of the inner structures of protein sequence pairs. In this section, we will show how to utilize the ADLDs to analyze the similarity/dissimilarity of a group of protein sequences.Generally, suppose that there areN protein sequences {Ψ1,Ψ2,…,ΨN}; then while applying the ADLDs to analyze the similarity/dissimilarity of these N sequences, the similarity/dissimilarity matrix of these N sequences can be obtained through the following steps.Step 1.
According to the method given in Section2.5, transform these N protein sequences into N numerical sequences {S1,S2,…,SN}.Step 2.
For a given protein sequence pair{Ψa,Ψb}, a∈{1,2,…,N}, b∈{1,2,…,N}, we can obtain their ADLD through adopting the method proposed in Section 2.6, and then we can obtain all of the SFs (including ASFs and BSFs) and FPs in the ADLD. Hence, we can obtain the lengths of these ASFs, the lengths of these BSFs, and the number of these FPs, respectively.Step 3.
Suppose that there are totallyL1 different ASFs such as {ASF1,ASF2,…,ASFL1}, L2 different BSFs such as {BSFl11,BSFl22,…,BSFlL2L2}, and L3 different FPs such as {FP1,FP2,…,FPL3} in the ADLD. And, in addition, for each ASFi and BSFljj, let their length be lengthi and lengthj, respectively, where i∈{1,2,…,L1} and j∈{1,2,…,L2}; then we can define thesimilarity degree (SD) of {Ψa,Ψb} as follows:
(17)SDΨa,Ψb=∑i=1L1lengthi+∑j=1L2lengthj+L3.
And, therefore, according to theseN protein sequences {Ψ1,Ψ2,…,ΨN}, we can obtain an N×Nmatching matrix (MM) as follows:
(18)MM=dijN×N,
where
(19)iiiiiiiiiiiiiiidij=SDΨi,Ψj,ifi≥j,0,else,fori∈1,2,…,N,j∈1,2,…,N.Step 4.
Based on the matching matrixMM and all of its components dij, where i∈{1,2,…,N} and j∈{1,2,…,N}, then we can obtain an N×Nsimilarity/dissimilarity matrix (SM) of these N protein sequences {Ψ1,Ψ2,…,ΨN} as follows:
(20)SM=sijN×N,
where
(21)iiiiiiiiiiiisij=1-Λdijdii,ifi≥j,0,else,iiiΛdijdii=1,ifdijdii≥1,dijdii,else,fori∈1,2,…,N,j∈1,2,…,N.According to the above steps, we present an example through implementing the ADLDs to analyze the similarity/dissimilarity of 16 ND5 proteins (illustrated in Table5) while letting δ=3 and illustrate the results of similarity/dissimilarity matrix in Table 6.Table 5
The basic information of 16 ND5 protein sequences.
Number
Name
Abbreviation
Access number
Length
1
Human
Human
ADT80430.1
603
2
Gorilla
Gorilla
NP_008222
603
3
Pigmy chimpanzee
Pi-chim
NP_008209
603
4
Common chimpanzee
C-chim
NP_008196
603
5
Fin-whale
Fin-whale
NP_006899
606
6
Blue-whale
Blue-whale
NP_007066
606
7
Rat
Rat
AP_004902.1
610
8
Mouse
Mouse
NP_904338
607
9
Opossum
Opossum
NP_007105
602
10
Sheep
Sheep
ABW22903.1
606
11
Goat
Goat
BAN59258.1
606
12
Lemur
Lemur
CAD13431.1
603
13
Cattle
Cattle
ADN11902.1
606
14
Hare
Hare
CAD13291.1
603
15
Gallus
Gallus
BAE16036.1
605
16
Rabbit
Rabbit
NP_007559.1
603Table 6
The similarity/dissimilarity matrix for the 16 ND5 proteins based on the ADLDs based method.
Human
Gorilla
Pi-chim
C-chim
Fin-whale
Blue-whale
rat
mouse
opossum
sheep
goat
lemur
cattle
hare
gallus
rabbit
Human
0.0000
Gorilla
0.1111
0.0000
Pi-chim
0.0720
0.0833
0.0000
C-chim
0.0814
0.0865
0.0510
0.0000
Fin-whale
0.3396
0.3285
0.3222
0.3301
0.0000
Blue-whale
0.3474
0.3333
0.3285
0.3301
0.0324
0.0000
Rat
0.3693
0.3622
0.3636
0.3716
0.3333
0.3381
0.0000
Mouse
0.3740
0.3686
0.3716
0.3748
0.3317
0.3333
0.1883
0.0000
Opossum
0.4476
0.4551
0.4290
0.4418
0.4515
0.4519
0.4513
0.4479
0.0000
Sheep
0.3020
0.2933
0.2871
0.2951
0.2023
0.2067
0.3149
0.3219
0.4121
0.0000
Goat
0.3036
0.2901
0.2871
0.2919
0.1958
0.2147
0.3166
0.3468
0.4202
0.0712
0.0000
Lemur
0.2989
0.2708
0.2839
0.2967
0.2557
0.2724
0.3166
0.3670
0.4055
0.2087
0.2317
0.0000
Cattle
0.3114
0.3045
0.3046
0.3062
0.1958
0.2051
0.3149
0.3173
0.4235
0.0906
0.1254
0.2184
0.0000
Hare
0.3146
0.3157
0.3062
0.3046
0.2832
0.2788
0.3166
0.3421
0.4023
0.2217
0.2508
0.2053
0.2532
0.0000
Gallus
0.4726
0.4920
0.4737
0.5008
0.4450
0.4423
0.4903
0.4743
0.4691
0.4239
0.4524
0.4680
0.4183
0.4660
0.0000
Rabbit
0.3255
0.3189
0.3222
0.3142
0.2896
0.2756
0.3084
0.3390
0.4332
0.2184
0.2603
0.2282
0.2612
0.0837
0.4434
0.0000Observing Table6, it is easy to find that there are some similar pairs such as (c-chim, pi-chim) with the distance0.0510, (human, c-chim) with the distance0.0814, (human, pi-chim) with the distance0.0720, (gorilla, c-chim) with the distance0.0865, (gorilla, pi-chim) with the distance0.0833, and (fin-whale, blue-whale) with the distance0.0324. And, among them, the opossum seems to be a peculiar mammal, since the shortest distance between it and the remaining mammals is more than0.4023. Obviously, the result is consistent with the fact that opossum is the most remote species from the remaining mammals.Additionally, gallus seems to be more peculiar than opossum, since the shortest distance between it and the remaining animals is more than0.4423, which is bigger than0.4023 (the shortest distance between Opossum and the remaining mammals). Obviously, the result is consistent with the fact that gallus is not a kind of mammal.Therefore, it is apparent that the results illustrated in Table6 are wholly consistent with the results of the known fact of evolution. That is to say, our ADLDs based method can be utilized as an effective way to analyze the similarities/dissimilarities of protein sequences.


### 3.2. The Phylogenetic Tree of the Protein Sequences Based on the ADLDs

Aphylogenetic tree is a diagram that is used to represent the evolutionary relationships of organisms that are thought to have a common ancestry, and it is a commonly used tool for researchers in some fields to help them analyze the clustering of different species.Obviously, only through observing the similarity/dissimilarity matrix illustrated in Table6, we will find that it is not very convenient to distinguish the similarity/dissimilarity of protein sequences. Therefore, in order to show the similarity/dissimilarity of the protein sequences more vividly and intuitively, according to the similarity/dissimilarity matrix illustrated in Table 6, then we will construct the phylogenetic tree of the above 16 ND5 proteins through adopting the software MEGA 6.06 that is provided by Tamura et al. [41], and the result is illustrated in Figure 3.Figure 3
The phylogenetic tree of the 16 species based on the ADLDs based method.From Figure3, it is obvious that we can not only find out the evolutionary relationships of these 16 ND5 protein sequences visually and intuitively but also know easily that the constructed phylogenetic tree is consistent with the results of the known fact of evolution to some degree.To further validate the performance of our ADLDs based method, we applied our method to analyze the similarity/dissimilarity of another group of proteins including 29 spike proteins of coronavirus and compared our method with the method proposed by Wen and Zhang [17] based on the above given 16 ND5 proteins and the following 29 spike proteins, respectively. The basic information of the 29 spike proteins is illustrated in Table 7.Table 7
The basic information of 29 spike proteins.
Number
Access number
Abbreviation
Length
1
CAB91145
TGEVG
1447
2
NP_058424
TGEV
1447
3
AAK38656
PEDVC
1383
4
NP_598310
PEDV
1383
5
NP_937950
HCoVOC43
1361
6
AAK83356
BCoVE
1363
7
AAL57308
BCoVL
1363
8
AAA66399
BCoVM
1363
9
AAL40400
BCoVQ
1363
10
AAB86819
MHVA
1324
11
YP_209233
MHVJHM
1376
12
AAF69334
MHVP
1321
13
AAF69344
MHVM
1324
14
AAP92675
IBVBJ
1169
15
AAS00080
IBVC
1169
16
NP_040831
IBV
1162
17
AAS10463
GD03T0013
1255
18
AAU93318
PC4127
1255
19
AAV49720
PC4137
1255
20
AAU93319
PC4205
1255
21
AAU04646
civet007
1255
22
AAU04649
civet010
1255
23
AAV91631
A022
1255
24
AAP51227
GD01
1255
25
AAS00003
GZ02
1255
26
AAP30030
BJ01
1255
27
AAP50485
FRA
1255
28
AAP41037
TOR2
1255
29
AAQ01597
TaiwanTC1
1255For the 29 spike proteins illustrated in Table7, we construct the phylogenetic tree in Figure 4. Since the spike protein sequences are very long (with more than 1100 amino acids), therefore, during simulation, we set δ=5 to avoid the effect of noise points.Figure 4
The phylogenetic tree of the 29 spike proteins of coronavirus constructed by adopting the ADLDs based method withδ=5.Generally,coronavirus can always be classified into four classes such as the Group I, the Group II, the Group III, and the SARS-CoVs (Severe Acute Respiratory Syndrome Coronaviruses). And, among these four classes, the Group I includes theCanine coronavirus (CCoV), theFeline coronavirus (FCoV), theHuman coronavirus 229E (HCoV-229E), thePorcine epidemic diarrhea virus (PEDV), and theTransmissible gastroenteritis virus (TGEV). The Group II includes theBovine coronavirus (BCoV),Human coronavirus OC43 (HCoV-OC43), theMurine coronavirus, Mouse hepatitis virus (MHV), the Porcine hemagglutinating encephalomyelitis virus (HEV), and theRat coronavirus (RtCoV). The Group III contains theAvian infectious bronchitis virus (IBV) and theTurkey coronavirus (TCoV).From observing Figure4, it is easy to know that the 29 spike proteins of coronavirus can be perfectly classified into the above four classes by our ADLDs based method.Finally, for the convenience of comparison, we illustrate the phylogenetic trees of the above given 29 spike proteins ofcoronavirus and 16 ND5 proteins, constructed by adopting the method proposed by Wen and Zhang [17], in Figures 5 and 6, respectively.Figure 5
The phylogenetic tree of the 29 spike proteins of coronavirus constructed by adopting the method proposed by Wen and Zhang.Figure 6
The phylogenetic tree of the 16 ND5 proteins constructed by adopting the method proposed by Wen and Zhang.Comparing Figure3 with Figure 6 and Figure 4 with Figure 5, respectively, it is obvious that the phylogenetic trees obtained by the method proposed by Wen and Zhang are quite unreasonable and not consistent with the known facts of evolution at all. But, on the contrary, the phylogenetic trees obtained by our ADLDs based method are not only quite reasonable but also consistent with the known facts of evolution to some degree. Therefore, there is no doubt that the performance of our method is much better than that of the method proposed by Wen and Zhang.


### 3.3. The Analysis of Intuition and Visuality of the ADLDs

In Section2.6, we have stated that the ADLDs of protein sequence pairs are intuitional and visual. In this section, we will further discuss the intuition and visuality of the ADLDs in detail.From Table6, we can obtain some similar pairs such as (fin-whale, blue-whale), (pi-chim, c-chim), (Human, c-chim), (cheep, goat), (human, pi-chim), and (hare, rabbit) and some dissimilar pairs such as (human, opossum) and (human, gallus), among the above given 16 ND5 proteins. From these similar/dissimilar pairs, we will choose three pairs including (human, gorilla), (human, opossum), and (human, gallus) as examples to further show the intuition and visuality of the ADLDs of these three protein sequence pairs. The ADLDs of these three similar/dissimilar pairs are illustrated in Figure 7, while letting δ=3.(a) The ADLD of the similar pair (human, gorilla); (b) the ADLD of the dissimilar pair (human, opossum); (c) the ADLD of the dissimilar pair (human, gallus).
(a)
(b)
(c)Observing Figure7, we can clearly find that the total length of all of the SFs in each of these three ADLDs satisfiesthe total length of all of the SFs in the ADLD of Figure7(a) > the total length of all of the SFs in the ADLD of Figure 7(b) > the total length of all of the SFs in the ADLD of Figure 7(c). Therefore, we can intuitively identify that the similarity of the proteins in each of these three protein sequence pairs satisfiesthe similarity of the proteins in the pair (human, gorilla) > the similarity of the proteins in the pair (human, opossum) > the similarity of the proteins in the pair (human, gallus).Moreover, from Figure7, we can also intuitively identify that the two protein sequences in the protein sequence pair (human, gorilla) are very similar to each other, since the total length of all of the SFs in the ADLD of Figure 7(a) looks very long. But, on the contrary, we can intuitively identify that the two protein sequences in either the protein sequence pair (human, opossum) or the protein sequence pair (human, gallus) are apparently dissimilar to each other, since both the total length of all of the SFs in the ADLD of Figure 7(b) and that in the ADLD of Figure 7(c) look very short.And, through statistic, we can know that the actual total lengths of all of the SFs in the ADLDs of these three protein sequence pairs (human, gorilla), (human, opossum), and (human, gallus) are 556, 288, and 248, respectively.Additionally, observing Figures2(a) and 2(b), hardly can we distinguish the total length of all of the SFs (including ASFs and BSFs) in the ADLD of Figure 2(a) and that in the ADLD of Figure 2(b), since the total lengths of all of the SFs in these two ADLDs look nearly the same. And, through statistic, we can know that the actual total lengths of all of the SFs in the ADLDs of Figures 2(a) and 2(b) are 123 and 120, respectively, and are really close to each other. But, through comparing Figure 2(a) with Figure 2(b) more carefully, we can further discover that, different from Figure 2(b), except for the SFs, there are also 6 different FPs in the ADLD of Figure 2(a), while there are no FPs in the ADLD of Figure 2(b); therefore, we can intuitively identify that the two protein sequences in the protein sequence pair (chimpanzee, human) are more similar to the two protein sequences in the protein sequence pair (human, gorilla).Hence, from the above descriptions, we can know that the ADLDs obtained by our newly proposed method are quite visual and intuitional and may be a powerful and effective tool for visual comparison of protein sequences and numerical sequences in other research fields.


## 3.1. Method for Similarity/Dissimilarity Analysis of Protein Sequences Based on the ADLDs

According to the above analysis, we have known that the ADLDs may be useful in analyzing the differences of the inner structures of protein sequence pairs. In this section, we will show how to utilize the ADLDs to analyze the similarity/dissimilarity of a group of protein sequences.Generally, suppose that there areN protein sequences {Ψ1,Ψ2,…,ΨN}; then while applying the ADLDs to analyze the similarity/dissimilarity of these N sequences, the similarity/dissimilarity matrix of these N sequences can be obtained through the following steps.Step 1.
According to the method given in Section2.5, transform these N protein sequences into N numerical sequences {S1,S2,…,SN}.Step 2.
For a given protein sequence pair{Ψa,Ψb}, a∈{1,2,…,N}, b∈{1,2,…,N}, we can obtain their ADLD through adopting the method proposed in Section 2.6, and then we can obtain all of the SFs (including ASFs and BSFs) and FPs in the ADLD. Hence, we can obtain the lengths of these ASFs, the lengths of these BSFs, and the number of these FPs, respectively.Step 3.
Suppose that there are totallyL1 different ASFs such as {ASF1,ASF2,…,ASFL1}, L2 different BSFs such as {BSFl11,BSFl22,…,BSFlL2L2}, and L3 different FPs such as {FP1,FP2,…,FPL3} in the ADLD. And, in addition, for each ASFi and BSFljj, let their length be lengthi and lengthj, respectively, where i∈{1,2,…,L1} and j∈{1,2,…,L2}; then we can define thesimilarity degree (SD) of {Ψa,Ψb} as follows:
(17)SDΨa,Ψb=∑i=1L1lengthi+∑j=1L2lengthj+L3.
And, therefore, according to theseN protein sequences {Ψ1,Ψ2,…,ΨN}, we can obtain an N×Nmatching matrix (MM) as follows:
(18)MM=dijN×N,
where
(19)iiiiiiiiiiiiiiidij=SDΨi,Ψj,ifi≥j,0,else,fori∈1,2,…,N,j∈1,2,…,N.Step 4.
Based on the matching matrixMM and all of its components dij, where i∈{1,2,…,N} and j∈{1,2,…,N}, then we can obtain an N×Nsimilarity/dissimilarity matrix (SM) of these N protein sequences {Ψ1,Ψ2,…,ΨN} as follows:
(20)SM=sijN×N,
where
(21)iiiiiiiiiiiisij=1-Λdijdii,ifi≥j,0,else,iiiΛdijdii=1,ifdijdii≥1,dijdii,else,fori∈1,2,…,N,j∈1,2,…,N.According to the above steps, we present an example through implementing the ADLDs to analyze the similarity/dissimilarity of 16 ND5 proteins (illustrated in Table5) while letting δ=3 and illustrate the results of similarity/dissimilarity matrix in Table 6.Table 5
The basic information of 16 ND5 protein sequences.
Number
Name
Abbreviation
Access number
Length
1
Human
Human
ADT80430.1
603
2
Gorilla
Gorilla
NP_008222
603
3
Pigmy chimpanzee
Pi-chim
NP_008209
603
4
Common chimpanzee
C-chim
NP_008196
603
5
Fin-whale
Fin-whale
NP_006899
606
6
Blue-whale
Blue-whale
NP_007066
606
7
Rat
Rat
AP_004902.1
610
8
Mouse
Mouse
NP_904338
607
9
Opossum
Opossum
NP_007105
602
10
Sheep
Sheep
ABW22903.1
606
11
Goat
Goat
BAN59258.1
606
12
Lemur
Lemur
CAD13431.1
603
13
Cattle
Cattle
ADN11902.1
606
14
Hare
Hare
CAD13291.1
603
15
Gallus
Gallus
BAE16036.1
605
16
Rabbit
Rabbit
NP_007559.1
603Table 6
The similarity/dissimilarity matrix for the 16 ND5 proteins based on the ADLDs based method.
Human
Gorilla
Pi-chim
C-chim
Fin-whale
Blue-whale
rat
mouse
opossum
sheep
goat
lemur
cattle
hare
gallus
rabbit
Human
0.0000
Gorilla
0.1111
0.0000
Pi-chim
0.0720
0.0833
0.0000
C-chim
0.0814
0.0865
0.0510
0.0000
Fin-whale
0.3396
0.3285
0.3222
0.3301
0.0000
Blue-whale
0.3474
0.3333
0.3285
0.3301
0.0324
0.0000
Rat
0.3693
0.3622
0.3636
0.3716
0.3333
0.3381
0.0000
Mouse
0.3740
0.3686
0.3716
0.3748
0.3317
0.3333
0.1883
0.0000
Opossum
0.4476
0.4551
0.4290
0.4418
0.4515
0.4519
0.4513
0.4479
0.0000
Sheep
0.3020
0.2933
0.2871
0.2951
0.2023
0.2067
0.3149
0.3219
0.4121
0.0000
Goat
0.3036
0.2901
0.2871
0.2919
0.1958
0.2147
0.3166
0.3468
0.4202
0.0712
0.0000
Lemur
0.2989
0.2708
0.2839
0.2967
0.2557
0.2724
0.3166
0.3670
0.4055
0.2087
0.2317
0.0000
Cattle
0.3114
0.3045
0.3046
0.3062
0.1958
0.2051
0.3149
0.3173
0.4235
0.0906
0.1254
0.2184
0.0000
Hare
0.3146
0.3157
0.3062
0.3046
0.2832
0.2788
0.3166
0.3421
0.4023
0.2217
0.2508
0.2053
0.2532
0.0000
Gallus
0.4726
0.4920
0.4737
0.5008
0.4450
0.4423
0.4903
0.4743
0.4691
0.4239
0.4524
0.4680
0.4183
0.4660
0.0000
Rabbit
0.3255
0.3189
0.3222
0.3142
0.2896
0.2756
0.3084
0.3390
0.4332
0.2184
0.2603
0.2282
0.2612
0.0837
0.4434
0.0000Observing Table6, it is easy to find that there are some similar pairs such as (c-chim, pi-chim) with the distance0.0510, (human, c-chim) with the distance0.0814, (human, pi-chim) with the distance0.0720, (gorilla, c-chim) with the distance0.0865, (gorilla, pi-chim) with the distance0.0833, and (fin-whale, blue-whale) with the distance0.0324. And, among them, the opossum seems to be a peculiar mammal, since the shortest distance between it and the remaining mammals is more than0.4023. Obviously, the result is consistent with the fact that opossum is the most remote species from the remaining mammals.Additionally, gallus seems to be more peculiar than opossum, since the shortest distance between it and the remaining animals is more than0.4423, which is bigger than0.4023 (the shortest distance between Opossum and the remaining mammals). Obviously, the result is consistent with the fact that gallus is not a kind of mammal.Therefore, it is apparent that the results illustrated in Table6 are wholly consistent with the results of the known fact of evolution. That is to say, our ADLDs based method can be utilized as an effective way to analyze the similarities/dissimilarities of protein sequences.


## 3.2. The Phylogenetic Tree of the Protein Sequences Based on the ADLDs

Aphylogenetic tree is a diagram that is used to represent the evolutionary relationships of organisms that are thought to have a common ancestry, and it is a commonly used tool for researchers in some fields to help them analyze the clustering of different species.Obviously, only through observing the similarity/dissimilarity matrix illustrated in Table6, we will find that it is not very convenient to distinguish the similarity/dissimilarity of protein sequences. Therefore, in order to show the similarity/dissimilarity of the protein sequences more vividly and intuitively, according to the similarity/dissimilarity matrix illustrated in Table 6, then we will construct the phylogenetic tree of the above 16 ND5 proteins through adopting the software MEGA 6.06 that is provided by Tamura et al. [41], and the result is illustrated in Figure 3.Figure 3
The phylogenetic tree of the 16 species based on the ADLDs based method.From Figure3, it is obvious that we can not only find out the evolutionary relationships of these 16 ND5 protein sequences visually and intuitively but also know easily that the constructed phylogenetic tree is consistent with the results of the known fact of evolution to some degree.To further validate the performance of our ADLDs based method, we applied our method to analyze the similarity/dissimilarity of another group of proteins including 29 spike proteins of coronavirus and compared our method with the method proposed by Wen and Zhang [17] based on the above given 16 ND5 proteins and the following 29 spike proteins, respectively. The basic information of the 29 spike proteins is illustrated in Table 7.Table 7
The basic information of 29 spike proteins.
Number
Access number
Abbreviation
Length
1
CAB91145
TGEVG
1447
2
NP_058424
TGEV
1447
3
AAK38656
PEDVC
1383
4
NP_598310
PEDV
1383
5
NP_937950
HCoVOC43
1361
6
AAK83356
BCoVE
1363
7
AAL57308
BCoVL
1363
8
AAA66399
BCoVM
1363
9
AAL40400
BCoVQ
1363
10
AAB86819
MHVA
1324
11
YP_209233
MHVJHM
1376
12
AAF69334
MHVP
1321
13
AAF69344
MHVM
1324
14
AAP92675
IBVBJ
1169
15
AAS00080
IBVC
1169
16
NP_040831
IBV
1162
17
AAS10463
GD03T0013
1255
18
AAU93318
PC4127
1255
19
AAV49720
PC4137
1255
20
AAU93319
PC4205
1255
21
AAU04646
civet007
1255
22
AAU04649
civet010
1255
23
AAV91631
A022
1255
24
AAP51227
GD01
1255
25
AAS00003
GZ02
1255
26
AAP30030
BJ01
1255
27
AAP50485
FRA
1255
28
AAP41037
TOR2
1255
29
AAQ01597
TaiwanTC1
1255For the 29 spike proteins illustrated in Table7, we construct the phylogenetic tree in Figure 4. Since the spike protein sequences are very long (with more than 1100 amino acids), therefore, during simulation, we set δ=5 to avoid the effect of noise points.Figure 4
The phylogenetic tree of the 29 spike proteins of coronavirus constructed by adopting the ADLDs based method withδ=5.Generally,coronavirus can always be classified into four classes such as the Group I, the Group II, the Group III, and the SARS-CoVs (Severe Acute Respiratory Syndrome Coronaviruses). And, among these four classes, the Group I includes theCanine coronavirus (CCoV), theFeline coronavirus (FCoV), theHuman coronavirus 229E (HCoV-229E), thePorcine epidemic diarrhea virus (PEDV), and theTransmissible gastroenteritis virus (TGEV). The Group II includes theBovine coronavirus (BCoV),Human coronavirus OC43 (HCoV-OC43), theMurine coronavirus, Mouse hepatitis virus (MHV), the Porcine hemagglutinating encephalomyelitis virus (HEV), and theRat coronavirus (RtCoV). The Group III contains theAvian infectious bronchitis virus (IBV) and theTurkey coronavirus (TCoV).From observing Figure4, it is easy to know that the 29 spike proteins of coronavirus can be perfectly classified into the above four classes by our ADLDs based method.Finally, for the convenience of comparison, we illustrate the phylogenetic trees of the above given 29 spike proteins ofcoronavirus and 16 ND5 proteins, constructed by adopting the method proposed by Wen and Zhang [17], in Figures 5 and 6, respectively.Figure 5
The phylogenetic tree of the 29 spike proteins of coronavirus constructed by adopting the method proposed by Wen and Zhang.Figure 6
The phylogenetic tree of the 16 ND5 proteins constructed by adopting the method proposed by Wen and Zhang.Comparing Figure3 with Figure 6 and Figure 4 with Figure 5, respectively, it is obvious that the phylogenetic trees obtained by the method proposed by Wen and Zhang are quite unreasonable and not consistent with the known facts of evolution at all. But, on the contrary, the phylogenetic trees obtained by our ADLDs based method are not only quite reasonable but also consistent with the known facts of evolution to some degree. Therefore, there is no doubt that the performance of our method is much better than that of the method proposed by Wen and Zhang.


## 3.3. The Analysis of Intuition and Visuality of the ADLDs

In Section2.6, we have stated that the ADLDs of protein sequence pairs are intuitional and visual. In this section, we will further discuss the intuition and visuality of the ADLDs in detail.From Table6, we can obtain some similar pairs such as (fin-whale, blue-whale), (pi-chim, c-chim), (Human, c-chim), (cheep, goat), (human, pi-chim), and (hare, rabbit) and some dissimilar pairs such as (human, opossum) and (human, gallus), among the above given 16 ND5 proteins. From these similar/dissimilar pairs, we will choose three pairs including (human, gorilla), (human, opossum), and (human, gallus) as examples to further show the intuition and visuality of the ADLDs of these three protein sequence pairs. The ADLDs of these three similar/dissimilar pairs are illustrated in Figure 7, while letting δ=3.(a) The ADLD of the similar pair (human, gorilla); (b) the ADLD of the dissimilar pair (human, opossum); (c) the ADLD of the dissimilar pair (human, gallus).
(a)
(b)
(c)Observing Figure7, we can clearly find that the total length of all of the SFs in each of these three ADLDs satisfiesthe total length of all of the SFs in the ADLD of Figure7(a) > the total length of all of the SFs in the ADLD of Figure 7(b) > the total length of all of the SFs in the ADLD of Figure 7(c). Therefore, we can intuitively identify that the similarity of the proteins in each of these three protein sequence pairs satisfiesthe similarity of the proteins in the pair (human, gorilla) > the similarity of the proteins in the pair (human, opossum) > the similarity of the proteins in the pair (human, gallus).Moreover, from Figure7, we can also intuitively identify that the two protein sequences in the protein sequence pair (human, gorilla) are very similar to each other, since the total length of all of the SFs in the ADLD of Figure 7(a) looks very long. But, on the contrary, we can intuitively identify that the two protein sequences in either the protein sequence pair (human, opossum) or the protein sequence pair (human, gallus) are apparently dissimilar to each other, since both the total length of all of the SFs in the ADLD of Figure 7(b) and that in the ADLD of Figure 7(c) look very short.And, through statistic, we can know that the actual total lengths of all of the SFs in the ADLDs of these three protein sequence pairs (human, gorilla), (human, opossum), and (human, gallus) are 556, 288, and 248, respectively.Additionally, observing Figures2(a) and 2(b), hardly can we distinguish the total length of all of the SFs (including ASFs and BSFs) in the ADLD of Figure 2(a) and that in the ADLD of Figure 2(b), since the total lengths of all of the SFs in these two ADLDs look nearly the same. And, through statistic, we can know that the actual total lengths of all of the SFs in the ADLDs of Figures 2(a) and 2(b) are 123 and 120, respectively, and are really close to each other. But, through comparing Figure 2(a) with Figure 2(b) more carefully, we can further discover that, different from Figure 2(b), except for the SFs, there are also 6 different FPs in the ADLD of Figure 2(a), while there are no FPs in the ADLD of Figure 2(b); therefore, we can intuitively identify that the two protein sequences in the protein sequence pair (chimpanzee, human) are more similar to the two protein sequences in the protein sequence pair (human, gorilla).Hence, from the above descriptions, we can know that the ADLDs obtained by our newly proposed method are quite visual and intuitional and may be a powerful and effective tool for visual comparison of protein sequences and numerical sequences in other research fields.


## 4. Conclusions

In this paper, a novel ADLDs based graphical representation of protein sequences is proposed, which is utilized to analyze the similarity/dissimilarity of protein sequences. To validate the performances of the new method, we select two groups of well-known protein sequences as examples, and, additionally, in order to observe the similarity/dissimilarity of protein sequences more intuitively, we construct the phylogenetic trees of protein sequences. The results show that our ADLDs based method not only has good performances and effects in the similarity/dissimilarity analysis of protein sequences but also does not require complex computation, since there are no high dimensional matrixes required. Therefore, it means that our ADLDs based method can work well in the analysis of protein sequences.



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*Source: 959753-2014-10-30.xml*