Evolutionary Relationship of Protein Sequences of SARS-CoV-2 and Other Viruses through Chaos Game Representation


Comparison between different biological sequences is a key step in bioinformatics when analyzing similarities of sequences and phylogenetic relationships. A method of graphically representing biological sequences known as Chaos Game Representation (CGR) has achieved many applications in the studies of bioinformatics. The key issue in the application of CGR is to extract as many useful features as possible from CGR. Initially, CGR was applied to DNA sequences, but in this paper, a CGR-based approach is used to extract suitable features for comparing protein sequences of SARS-CoV-2 and other viruses. For this aim, several viral protein sequences from 12 groups are considered and CGR centroid, amino acid frequency, compounded frequency, Shannon entropy, and Kullback-Lieber Discrimination Information are applied to find the inter-relationship among the sequences. The experimental results demonstrate the potential strengths of CGR-based method for examining the evolutionary relationship of protein sequences. Our method is powerful for extracting effective features from protein sequences, and therefore important in classifying proteins and inferring the phylogeny of viruses.

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Hill, M. , Simmons, K. and Sengupta, D. (2022) Evolutionary Relationship of Protein Sequences of SARS-CoV-2 and Other Viruses through Chaos Game Representation. Computational Molecular Bioscience, 12, 123-143. doi: 10.4236/cmb.2022.123008.

1. Introduction

Proteins are complex molecules that play a critical role in several functions of the body as well as the structure of tissue and organs. They are comprised of amino acids which are connected in long chains ranging from a few hundred to several thousand depending on the protein. These chains of amino acids determine the structure and function of a protein, which include the transport and storage of structural components, and enzymes. By studying the structure and function of proteins, we can hurdle some of the obstacles in understanding the evolutionary relationships of organisms. The 20 amino acids are Alanine (A), Arginine (R), Asparagine (N), Aspartic Acid (D), Cysteine (C), Glutamic acid (E), Glutamine(Q), Glycine (G), Histidine (H), Isoleucine (I), Leucine (L), Lysine (K), Methionine (M), Phenylalanine (F), Proline (P), Serine (S), Threonine (T), Tryptophan (W), Tyrosine (Y), and Valine (V) [1]. Each amino acid has certain physical and chemical properties which distinguish it from others. In general, the biological function of a protein is determined by its 3-dimensional structure which is dependent on the linear sequence of amino acids. Rigden [2] presented that one of the fundamental principles of molecular biology is that proteins having similar sequences possess similar functions. This leads to difficulty when comparing closely and distantly related sequences. Similarity analysis of protein sequences plays an important role in protein sequence studies, e.g. the prediction or classification of protein structures and functions. In recent years, many numerical representation methods have been proposed and then applied in protein classification.

Apart from representing biological sequences into numerical expression directly, many other numerical representations are constructed by first giving the sequence a graphical representation and then studying the image numerically [3]. Chaos game representation (CGR) was originally applied to bioinformatics as an image representation of DNA sequences by Jefferey in 1990 [4]. The four nucleotides {A, T, G, C} were put on 4 vertices of the unit square, and every DNA sequence was mapped to a series of points inside the unit square in 2-dimensional space. Being capable of discovering the inner pattern of gene sequences, CGR has been widely used in the investigation of DNA sequences in [5] - [11]. Encouraged by the CGR of DNA sequences, the CGR of protein sequences has also been extensively studied by many researchers. While DNAs are composed of four kinds of nucleotides, proteins are made up of twenty kinds of amino acids. Thus, it remains to decide the distribution of the 20 amino acids when promoting CGR to the image representation of proteins.

Fiser [12] was one of the first to find a method to improve such techniques by creating a 20-sided polygon with each vertex representing one of the 20 amino acids. Another representation of the 20 amino acids was applied by Randic [13] in which the CGR exists within the unit circle. This approach ordered the amino acids alphabetically in comparison to organization based on their physiochemical properties. The properties of the amino acids serve as vital information for the characterization of protein sequences and this was noted by Randic. Considering the limitation that a 20-vertex CGR cannot be used to demonstrate the similarity of protein sequences with conservative substitution, Basu [14] proposed a 12-vertex CGR, with each vertex of a regular 12-sided polygon representing an amino acid with its conservative substitutions. The number of the vertices in CGR was then reduced to four [15] [16], with each vertex of the square representing one of the four groups of amino acids, that is, the non-polar, uncharged polar, negative polar, and positive polar groups. The reduction in the vertices of CGR images can help represent the similarity in protein sequences.

Up to now, CGR method has achieved many applications in the studies of bioinformatics. The key issue in the application of CGR is to extract as many useful features as possible from CGR and several studies showed that those extracted features play important roles in protein studies [17] - [23]. One of the most used feature extraction methods is the so-called FCGR, in which the CGR image is split into small grids and the frequencies of points falling into each grid are taken as the feature of the corresponding protein sequence. In our previous work [24], we used FCGR to study the similarity of coronavirus sequences. While FCGR has been used mainly for coronavirus genome sequence encoding and classification, we modified it in this study to work also for protein sequences.

For this study, HTLV 1, HIV 1, HIV 2, Ebola, Dengue, Middle Eastern Respiratory Syndrome (MERS), Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV), and Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) were used for protein sequence comparison. SARS-CoV-2 has been detrimental to the human population over the past year. At the time of this report, more than 179 million people have contracted the virus and over 3.8 million of those have been fatal. The first pathogenic novel coronavirus, discovered in 2003 and named SARS-CoV, caused SARS, serious and atypical pneumonia. The second, MERS-CoV, emerged a decade later in the Middle East and caused a similar respiratory ailment called Middle East respiratory syndrome (MERS). Since its identification, 2494 cases of MERS-CoV infection and nearly 900 deaths have been documented. The SARS-CoV epidemic proved larger but less deadly, with approximately 8000 cases and nearly 800 deaths. There are other four coronaviruses that cause colds in humans-known as HCoV-229E, HCoV-NL63, HCoV-OC43, and HCoV-HKU1 [24]. SARS-CoV-2 is the third pathogenic novel coronavirus. Identifying ways to better understand such viruses is of grave importance to the human population. Such major outbreaks demand classification and origin of the virus genome sequence, for planning, containment, and treatment. Motivated by the above need, we report a method combining with CGR to perform clustering analysis and create a phylogenetic tree based on it.

For this report, CGR is used for the identification of several hundred protein sequences into their respective viral groups through feature extraction. These features include CGR centroid, amino acid frequency, compounded frequency, Shannon entropy, and Kullback-Lieber Discrimination Information. Due to the scale independence of CGR, smaller components of the CGR graph can be used to help explain the bigger picture. This points to the potential of extracting smaller features of the graph and using them to better explain the protein sequence as a whole. After the application of our proposed method, we apply multidimensional scaling (MDS) to the data. With this 2D and 3D projections of the data can be obtained for clustering analysis. Kruskal [25] first introduced this method of information visualization which takes the distance matrices computed from our methods as input. In turn, a representation of each viral sequence is created in euclidean space with corresponding distances between sequences that are equivalent to their distance given in the matrix. Therefore, similar viral sequences should be relatively close in this representation which has been previously shown for DNA sequences [26] [27].

2. Methods

In this section, we describe the dataset used for our experiments, then discuss the proteins version of CGR, give an overview of the three main steps of our experiments, and conclude with a description of features that we considered.

2.1. Dataset

Data acquisition: All protein sequences were downloaded in FASTA format from the database for our analysis: NCBI (https://www.ncbi.nlm.nih.gov/). The data sets shown in Tables 1-6 are the accession numbers of 510 viral strains in 12 groups that we used for our experiments.

The HIV_1 group consisted of Gag-pol, Gag-pol poly, Gag-pol fusion, and Gag-pol fusion poly proteins. For HIV_2, pol, pol poly, Gag-pol poly, and Gag-pol fusion poly proteins, the Dengue group consisted of only polyproteins. HTLV group contained pol, polymerase, Gag/pol precursor, Gag-pro-pol poly, and Gag-protease proteins. SARS_CoV and SARS_CoV-2 encompassed six groups, two for ORF1a polyprotein, and two for ORF1ab polyprotein, one spike glycoprotein and one surface glycoprotein. MERS group contained 1a poly, 1ab poly, ORF1a, ORF1ab, OR1ab poly, and replicase poly proteins. Lastly, the Ebola group consisted of RNA-dependent RNA polymerase, RNA-directed RNA polymerase, polymerase, and L proteins.

2.2. CGR of Proteins

CGR is an algorithm that uses iterations in order to generate a pattern by utilizing the nucleotides in DNA or amino acids in protein sequences. CGR assigns a coordinate value to each alphabet in a sequence and hence a characteristic visual pattern is generated for each sequence. In the case of a DNA sequences, CGR assigns each of the four possible nucleotides A, T, G and C to one of the four vertices of a square. In our study, we used protein sequences; the 20 amino acids were divided into 4 groups, and each of these groups (designated A, B, C and D) was assigned to one of the four vertices of the square. We used groups based on amino acid residue chemical properties (charge and polarity): A = D, E (negatively charged); B = K, R, H (positively charged); C = S, T, N, C, Y, Q (neutral/polar); D = G, A, V, L, I, M, P (neutral/nonpolar).

Let the vertices of the unit square be: A = ( 0,0 ) , B = ( 0,1 ) , C = ( 1,1 ) , and D = ( 1,0 ) . Successive points in the CGR were generated by an iterated function system defined by the following formula

( x i + 1 , y i + 1 ) = ( x i + T x ( i ) 2 , y i + T y ( i ) 2 )

Table 1. HIV_1 & HIV_2 data sets.

Table 2. SARS_CoV & SARS_CoV-2 ORF1ab polyprotein data sets.

Table 3. MERS & Dengue data sets.

Table 4. Ebola & HTLV data sets.

Table 5. SARS_CoV & SARS_CoV-2 ORF1a polyprotein data sets.

Table 6. SARS_CoV spike glycoprotein & SARS_CoV-2 surface glycoprotein data sets.

where T x ( i ) is the x coordinate and T y ( i ) is the y coordinate of the vertex of the corresponding group of the next amino acid in the sequence. To create a CGR image, we first began with an initial point ( 0.5,0.5 ) , the center of a unit square in quadrant 1 of the xy-plane. A point is plotted half the distance from this vertex and the previous coordinate. The output file contained x, y coordinate values for each amino acid present in the input sequence. These x and y coordinate values were plotted as scatterplots. Some examples of the CGR of several viruses used in this report are shown in Figures 1-5.

2.3. Overview

The method we used to analyze and classify protein sequences has three steps: 1) generate graphical representations (images) of each Protein sequence using Chaos Game Representation (CGR), 2) compute all pairwise distances between these images using one of the following features, and 3) visualize the interrelationships implied by these distances as two- or three-dimensional maps, using Multi-Dimensional Scaling (MDS).

2.4. CGR Centroid

Once the CGR is created for a protein sequence, the CGR square is divided into four cells. Each cell represents one of the four groups, { A i , B i , C i , D i ; i = 1 , 2 , , n } where n is the length of the sequence. These cells correspond to the vertex located in that cell. The points in each cell are then averaged to find the centroid of each cell denoted by

Figure 1. CGR of Dengue (left) and Ebola (right).

Figure 2. CGR of HTLV (left) and MERS (right).

Figure 3. CGR of HIV_1 (left) and HIV_2 (right).

Figure 4. CGR of SARS_CoV spike glyco (left) and SARS_CoV ORF1a (right).

Figure 5. CGR of SARS_CoV-2 surface glyco (left) and SARS_CoV-2 ORF1a (right).

C k = ( i = 1 n a i ( x ) n , j = 1 n a j ( y ) n )

where a i ( x ) and a i ( y ) are the x and y coordinates respectively in a cell and k = 1 , 2 , 3 , 4 . This gives four centroids C 1 , C 2 , C 3 and C 4 for comparison of viral sequences.

2.5. CGR Centroid Bisection

Upon calculation of the four CGR centroids, a rectangle is created from these vertices. Next, the diagonals of this rectangle are constructed and their intersection is taken as the CGR Centroid Bisection denoted B C ( x ) of viral sequence x.

B C ( x ) = C 1 + C 4 2

2.6. Amino Acid Frequency

The next method of sequence comparison examined is the amino acid frequency (AAF) of 2mers. A 2mer is subsequence of length 2 of a string of characters and they are found by taking the cross product between the set of amino acids and itself. This yields 20 2 = 400 possible 2mers and some of these include: DE, MA, AR, HE, and RT. The frequency of each 2mer is calculated as follows

p i j = Numberofoccurencesof2mer 400

1 i j 400 . Several distance measures can then be obtained by comparing the amino acid FCGR of viral sequences. One distance metric that encompasses two others is the minkowski distance and is derived as follows

i = 1 n ( | p i j p i j | t ) 1 t

Note that when t = 1, we have

M = i = 1 n ( | p i j p i j | )

which is manhattan distance and when t = 2, we have

E = i = 1 n ( | p i j p i j | 2 )

euclidean distance.

2.7. Group Frequency Chaos Game Representation

Each cell in the CGR of protein contains an x amount of points and by dividing this amount by four for the four cells; we have the group frequency chaos game representation (GFCGR). With this the frequencies are defined for the four groups of amino acids as opposed to just one. The GFCGR is defined as follows:

GFCGR ( z ) = Numberofoccurencesofaminoacidinagroup z Lengthofthesequence

where z = { A , B , C , D } .

2.8. Kullback-Leibler Discrimination Information

A previous method introduced by Li [20] utilized the Kullback-Leibler Discrimination Information for sequence comparison. This comparison proved useful and in this report we further extend this method to be applicable with our previously mentioned methods. Given a discrete random variable Y, different distribution laws can be applied. For example under Hypothesis 1, we have

( Y p 1 ( y ) ) = ( y 1 y 2 y n p 1 ( y 1 ) p 1 ( y 2 ) p 1 ( y n ) )

Under Hypothesis 2, we have

( Y p 2 ( y ) ) = ( y 1 y 2 y n p 2 ( y 1 ) p 2 ( y 2 ) p 2 ( y n ) )

These distributions can be compared by using the Kullback-Leibler Discrimination Information denoted by

I ( p 1 , p 2 ) = i = 1 n p 1 ( a i ) log p 1 ( a i ) p 2 ( a i )

In this report, we let these distributions be the 2mer AAF of viral genomes. So for viruses x and y, we have I ( x , y ) , but due to its directed divergence I ( x , y ) might not necessarily equal I ( y , x ) . For this reason, the metric J ( a , b ) is defined as follows

J ( x , y ) = I ( x , y ) + I ( y , x )

Note that when x = y , J ( x , y ) = 0 . We also note that for any two viral sequences x and y, J ( x , y ) = J ( y , x ) . Li [20] noted that this method can accurately measure the dissimilarity between two sequences.

2.9. Compounded Frequency

Another method for sequence comparison that has been previously examined is the compounded frequency. This method was proposed by Almeida [3] for comparison of biological sequences. First we denote the compounded frequency nw as follows

n w = i = 1 k x i y i

The compounded frequency is then used in conjunction with the Pearson correlation coefficient, rw for sequence comparison.

r w = i = 1 k x i μ x s x y i μ y s y x i y i n w


s x = i = 1 k ( x i μ x ) 2 x i y i n w


s y = i = 1 k ( y i μ u ) 2 x i y i n w

with μ x and μ y defined as follows

μ x = i = 1 k x i 2 y i n w

μ y = i = 1 k y i 2 x i n w

Previous studies used this method for comparison of the FCGR of two sequences. Similarly, we use the 2mer AAF to find the rw between two sequences. By using the weight of nw, each 2mer is proportional to its frequency. Now we define the sequence distance as d = 1 r w , which has values from 0 - 2. For d > 1 , a negative correlation exists and for d < 1 a positive correlation exists. When d = 0 , the sequences are exactly similar.

2.10. Shannon Entropy

The Shannon information index has been used in some of our past work as well as other studies. It is denoted

S 2 = i = 1 k p i log 2 ( p i ) = i = 1 k p i log 2 ( 1 p i )

where 2merAAF = p 1 , p 2 , , p n , 1 i n . This method has been used in some of our past works for sequence comparison. In this report we use this method as a measure of the amount of information contained within a sequence of proteins.

3. Results

For the Shannon entropy, 2mer AAF and GFCGR the manhattan distance is used. The euclidean distance is applied to both the CGR centroids and CGR centroid bisections, while J ( x , y ) and Pearson correlation have the respective distance measures. MDS is then applied to the distance matrices to create 2D and 3D projections shown in Figures 6-12.

To rank the effectiveness of each distance metric, we define the δ ( x , y ) function as in [26] of two viral sequences x and y as follows

δ ( x , y ) = ( 0 ,if x and y belongtosameviralgroup 1 ,otherwise

Figure 6. 2mer AAF 2D and 3D MDS charts.

Figure 7. CGR Centroid Bisection 2D and 3D MDS charts.

Figure 8. CGR Centroid 2D and 3D MDS charts.

Figure 9. Compounded Freq 2D and 3D MDS charts.

Figure 10. GFCGR 2D and 3D MDS charts.

Figure 11. Kullback-Leibler 2D and 3D MDS charts.

Figure 12. Shannon Entropy 2D and 3D MDS charts.

With this function we create a 400 × 400 distance matrix of the viruses and take the upper triangular matrix as a vector U δ . Next, we take the upper triangle matrix, U α , α 2merAAF , J ( x , y ) , S 2 , D = 1 r w , GFCGR, CGR Centroid, CGR Centroid Bisection of each of the 7 distance matrices for comparison with U δ . The Pearson correlation coefficient is used to establish how well a distance measure fits a particular viral sequence to its corresponding group cluster. We denote this coefficient as

P α = σ α δ σ α σ δ

with a range of [ 1,1 ] . Values of 1 indicate a linear correlation between U δ and U α while a value of 0 indicates the pair is unrelated. The values of P α for each distance measure are shown in Table 7.

We see that of the distance measures, 2mer AAF is most closely related with U δ . Further confirmation of this is shown in the 2D and 3D MDS charts for 2mer AAF Figure 6, which show a good separation of the viral sequences into their respective groups. It can also be noted that viruses belonging to the coronavirus family cluster close together as do viruses belonging to the HIV family. We expect this as these viruses are more closely related than say HTLV or Dengue. In fact, SARS_CoV ORF1a and SARS_CoV-2 ORF1a overlap as do SARS_CoV ORF1ab and SARS_CoV-2 ORF1ab. This is indicative of a distance measure of almost 0, which shows just how closely related they are. Other measures such as Shannon entropy and GFCGR which have the lowest correlation with U δ , P α = 0.147915 and 0.36562 respectively, show a lack of separation between viral groups in their MDS charts Figure 10 and Figure 12.

4. Discussion and Conclusions

Feature extractions of protein sequences play an important role in protein sequence similarity studies. Although many methods have been proposed for extracting features of protein sequences, most of them showed great limits in practical applications. Many studies have shown that the CGR-based strategy would be one of the most useful approaches for protein feature extractions, and the so-called FCGR method is currently the most frequently used method-based CGR, however, a large amount of useful information, e.g. physicochemical properties of amino acids and the distribution information of points in the CGR image were not taken into consideration in the method of FCGR.

In this study, CGR was used for the identification of several hundred protein sequences into their respective viral groups through feature extraction. These features include CGR centroid, amino acid frequency, compounded frequency, Shannon entropy, and Kullback-Lieber Discrimination Information.

The method, we used to analyze and classify protein sequences, has three steps: 1) generate graphical representations (images) of each Protein sequence using Chaos Game Representation (CGR), 2) compute all pairwise distances between these images, and 3) visualize the interrelationships implied by these distances as two- or three-dimensional maps, using Multi-Dimensional Scaling (MDS).

Table 7. P α of distance metrics.

Figure 13. Dendrogram made using 2mer AAF of SARS_CoV-2 ORFlab.

Figure 14. Phylogenetic tree of SARS_CoV-2 ORF1ab from NCBI website.

Several distance metrics were introduced for comparison as well as a method of ranking these metrics. Our quantitative comparison of seven different distances suggests that the Kullback-Lieber Discrimination Information as well as the manhattan distance of 2mer AAF outperform all other distances. Our findings suggest that the Kullback-Lieber Discrimination Information as well as the manhattan distance of 2mer AAF is best in clustering viruses into their respective groups. This shows the importance of the frequency of 2mers in correctly identifying viral sequences. We compare the results of the phylogenetic tree of SARS_CoV-2 ORF1ab obtained from our 2mer AAF distance method with those given in the NCBI site in Figure 13 and Figure 14. The NCBI method performs equally well with our 2mer distance method. The two-dimensional and three-dimensional Molecular Distance Maps we obtain, which visualize the simultaneous interrelationships among the sequences in our dataset, show this method’s potential. Further analysis is needed to explore this method’s potential for the analysis of closely related sequences.

In conclusion, our distance comparison results on datasets illustrate the potential strengths of CGR-based method for examining the evolutionary relationship. Our method is powerful for extracting effective features from protein sequences, and therefore important in classifying proteins and inferring the phylogeny of viruses.


This work was done while D.C.S. mentored undergraduate student Kevin Simmons and a graduate student Matthew Hill. Kevin Simmons was funded by NIH-Minority Access to Research Career (MARC) Program Grant # NIH-NIGMS T34 GM 100831-09. This research was also partially funded by the University of North Carolina System’s Covid-19-related grant. The authors also would like to thank Mr. Joel W. Perry for spending time to fix all the Latex errors.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.


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