Correlated Mutations of Positions Among Structural Proteins in Delta

and Omicron Variants for SARS-CoV-2 Amino Acid Sequences

Yuichi Shimaya and Kouich Hirata

Kyushu Institute of Technology, Kawazu 680-4, Iizuka 820-8502, Japan

Keywords:

Correlated Mutation, Structural Proteins, Spike Protein Substutution, Amino Acid Sequences, SARS-CoV-2,

Delta Variant, Omicron Variant.

Abstract:

In this paper, we ﬁnd the correlated mutations of positions among structural proteins of spike, envelop, mem-

brane and nucleocapsid proteins in amino acid sequences of SARS-CoV-2. Here, we adopt the algorithm de-

signed by Shimada et al. (2012) of ﬁnding the correlated mutations formulated by joint entropy. In particular,

we discuss whether or not the found correlated mutations contains spike protein substitutions in SARS-CoV-2

Delta and Omicron variants.

1 INTRODUCTION

Coronavirus disease 19 (COVID-19) is caused by a

novel coronavirus designated as severe acute respi-

ratory syndrome coronavirus 2 (SARS-CoV-2). The

SARS-CoV-2 is an enveloped virus with a positive-

sense, single-stranded RNA genome. Figure 1 illus-

trates the schematic presentation of the SARS-CoV-2

genome organization, the canonical subgenomic mR-

NAs and the virion structure (Kim et al., 2020).

Coronaviruses (CoVs) were named because they

are characterized by spike protein projections on the

surface of the viral particles, and their shape resem-

bles a crown (corona) under electron microscopy, the

virion structure in Figure 1.

The CoVs carry the largest genomes among all

RNA virus families. The genomic RNA is trans-

lated to produce nonstructural protein (nsps) from two

open reading frames (ORFs), ORF1a and ORF1b.

ORF1a and ORF1b encode 11 and 5 nonstructural

proteins; nsp1 to nsp11 and nsp12 to nsp16, respec-

tively. Whereas ORF1a is translated directly from the

genomic RNA, the expression of ORF1b requires a -1

rebosomal frameshift near the end of ORF1, resulting

in a single ORF1ab polypeptide.

Downstream from the ORF1ab, there exist

ORFs encoding a few or more than 10 struc-

tural/nonstructural proteins. The common structural

proteins of CoVs are spike (S), envelope (E), mem-

brane (M) and nucleocapsid (N) proteins. SARS-

CoV-2 is also known to have at least 6 accessory

proteins, ORF3a, ORF6, ORF7a, ORF7b, ORF8 and

ORF10. However, the ORFs have not yet been exper-

imentally veriﬁed for expression. Therefore, it is cur-

rently unclear which accessory genes are actually ex-

pressed from this compact genome (Kim et al., 2020).

On the other hand, since the correlated muta-

tions of positions in amino acid sequences are fre-

quently observed among spatially close residues and

valuable for analyzing the structure of proteins, many

researchers (cf. (Jeong and Kim, 2010; Lee and

Kim, 2009; Oliveria et al., 2002)) have developed to

ﬁnd them. In their works, the correlated mutations

are formulated by using an entropy (Oliveria et al.,

2002), a CM-score (Lee and Kim, 2009) or log-odds

scores (Jeong and Kim, 2010).

In this paper, we adopt the correlated mutations

of positions based on a joint entropy ratio introduced

by Shimada et al. (Shimada et al., 2012). Here, the

joint entropy ratio of positions is the ratio of the mini-

mum entropy in each position in them to the joint en-

tropy (cf., (Cover and Thomas, 2006)) of them. Then,

we say that positions are correlated (Shimada et al.,

2012) if the joint entropy ratio of them is greater than

a given joint entropy threshold τ (0 < τ ≤ 1). Further-

more, based on their correlated mutations, they have

designed the algorithm FINDCM

of ﬁnding all of the

correlated mutations with the joint entropy threshold

τ and an exclusion threshold ρ (ρ > 0) to exclude the

positions in which elements do not change well.

Recently, Yonashiro et al. (Yonashiro et al., 2022)

have applied the algorithm FINDCM to ﬁnd the corre-

lated mutations of positions for structural proteins of

The algorithm FINDCM in this paper coincides with

an algorithm named by FINDCM2 in (Shimada et al.,

2012).

344

Shimaya, Y. and Hirata, K.

Correlated Mutations of Positions Among Structural Proteins in Delta and Omicron Variants for SARS-CoV-2 Amino Acid Sequences.

DOI: 10.5220/0011676000003411

In Proceedings of the 12th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2023), pages 344-349

ISBN: 978-989-758-626-2; ISSN: 2184-4313

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

Figure 1: Schematic presentation of the SARS-CoV-2 genome organization, the canonical subgenomic mRNAs and the virion

structure (Kim et al., 2020).

S, E, M and N proteins from amino acid sequences of

SARS-CoV-2, by focusing on the positions observed

as spike protein substitutions for SARS-CoV-2 Delta

variant reported from CDC

and NIIB

Here, SARS-CoV-2 Delta and Omicron variants

are classiﬁed to VOCs (Variant of Concerns) from

June in 2021 to April in 2022 and from November in

2021 to today, respectively, from CDC. Here, a VOC

is a variant for which there is evidence an increase in

transmissibility, more several disease, signiﬁcant re-

duction in neutralization by antibodies generated dur-

ing previous infection or vaccination, reduced effec-

tiveness of treatments or vaccines, or diagnostic de-

tection failures

Their variants are characterized as the spike pro-

tein substitutions. Table 1 illustrates the spike pro-

tein substitutions for SARS-CoV-2 Delta and Omi-

cron variants. Here, a

denotes that an amino acid

at the position n is substituted to another amino

acid a

. Also “(*)” denotes the substitution detected

in some sequences but not all. Furthermore, “del211

“ denotes that the amino acid at the position 211 is

deleted, “del69-70” denotes that the amino acids from

the position 69 to 70 are deleted and “ins214EPE” de-

notes that the amino acids EPE are inserted from the

position 214 to 215.

Hence, in this paper, after ﬁnding the correlated

mutations obtained by the algorithm FINDCM under

τ and ρ, we discuss whether or not the found corre-

lated mutations among all of the structural proteins

contains spike protein substitutions in not only SARS-

CoV-2 Delta variant and but also SARS-CoV-2 Omi-

cron variant. Note that the results for the Delta variant

are updated from the previous work (Yonashiro et al.,

2022) by using newly amino acid sequences.

CDC, Centers for Disease Control and Prevention.

https://www.cdc.gov.

NIIB, National Institute of Infectious Diseases.

https://www.niid.go.jp/nidd/ja.

Table 1: The spike protein substitutions for SARS-CoV-2

Delta and Omicron variants.

Delta variant

T19R (V70F*) T95I G142D del156

del157 R158G (A222V*) (W258L*) (K417N*)

L452R T478K D614G P681R D950N

Omicron variant

A67V del69-70 T95I del142-144 Y145D

del211 L212I ins214EPE G339D S371L

S373P S375F K417N N440K G446S

S477N T478K E484A Q493R G496S

Q498R N501Y Y505H T547K D614G

H655Y N679K P681H N764K D796Y

N856K Q954H N969K L981F

2 FINDING CORRELATED

MUTATION

In this section, we introduce the algorithm FINDCM

Let Σ be an alphabet and Σ

the set of all strings

on Σ with length n. Also let [n] be {1,..., n}. For

a string w ∈ Σ

and a set I = {i

,... ,i

} ⊆ [n] (1 ≤

< ·· · < i

≤ n), we denote the string w[i

]··· w[i

]

constructed from concatenating the symbols at from

to i

by w[I]. Furthermore, let P(w[I] = v) be the

probability that w[I] is v ∈ Σ

Deﬁnition 1. Let S ⊆ Σ

and I ⊆ [n]. Then, we deﬁne

the joint entropy H

(I) of S at a set I of positions (cf.,

(Cover and Thomas, 2006)) as:

(I) = −

∑

v∈Σ

|I|

,w∈S



P(w[I] = v) × logP(w[I] = v)



An entropy of S at a position i ∈ [n] coincides with

({i}) (cf., (Cover and Thomas, 2006)), which we

denote by H

(i). Then, we deﬁne the joint entropy

ratio R

(I) of S at a set I of positions as:

(I) =

min{H

(i) | i ∈ I}

(I)

Correlated Mutations of Positions Among Structural Proteins in Delta and Omicron Variants for SARS-CoV-2 Amino Acid Sequences

345

For the joint entropy ratio, the following lemma

holds.

Lemma 1. (Shimada et al., 2012) Let S ⊆ Σ

and

I, J ⊆ [n]. If I ⊆ J, then H

(I) ≤ H

(J).

Then, we formulate the correlated mutation by in-

troducing the joint entropy threshold as the main topic

in this paper.

Deﬁnition 2. (Shimada et al., 2012) Let S ⊆ Σ

and

I ⊆ [n]. Also let 0 < τ ≤ 1 be a joint entropy thresh-

old. Then, we say that I is correlated if R

(I) ≥ τ.

Furthermore, we call the mutations at I in S corre-

lated mutations of S.

When setting τ to 1, that is, R

(I) = 1, the corre-

lated mutations are exact. On the other hand, when

setting τ less than 1, that is, R

(I) < 1, the correlated

mutations contains exception. Furthermore, we use

another threshold ρ ≥ 0, called an exclusion thresh-

old, to exclude the positions in which elements do not

change well.

The algorithm FINDCM (Shimada et al., 2012) in

Algorithm 1, which is based on the set enumeration

algorithm (Rymon, 1992), describes the algorithm of

ﬁnding all of the correlated mutations with an entropy

pruning at lines from 5 to 7. Here, the correctness of

the entropy pruning follows from the following theo-

rem.

Theorem 1. (Shimada et al., 2012) Let S ⊆ Σ

and

I ⊆ [n]. If R

(I) ≥ τ, then it holds that τH

(i) ≤

( j) ≤ H

(i)/τ for every i, j ∈ I.

3 EXPERIMENTAL RESULTS

In this section, by focusing on the structural proteins,

that is, spike (S), envelope (E), membrane (M) and

nucleocapsid (N) proteins, we give experimental re-

sults of ﬁnding correlated mutations among them.

We use amino acid sequences of the four structural

proteins for a Delta variant and an Omicron variant

provided from NCBI

, whose length of amino acid

sequences for S, E, M and N proteins is 1,273, 75,

222 and 419, respectively (Nakagawa and Miyazaki,

2020). The amino acid sequences of the Delta variant

are stored from April 1 in 2021 to May 31 in 2022

and the number of sequences is 834,870. On the other

hand, the amino acid sequences of the Omicron vari-

ant are stored from October 1 in 2021 to July 29 in

2022 and the number of sequences is 393,655.

We use amino acids as 20 characters of A, C, D, E,

F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W and Y, so the

NCBI, National Center for Biotechnology Informa-

tion. https://www.ncbi.nlm.nih.gov.

procedure FINDCM(S,τ,ρ)

/* S ⊆ Σ

, 0 < τ ≤ 1, ρ ≥ 0 */

C ← {1, . ..,n};1

for i = 1 to n do2

if H

(i) ≥ ρ then3

C ← C \ {i}; C

← C;4

foreach j ∈ C

do5

if H

( j) ≥ ρ and



( j) < τH

(i)

or H

(i)/τ < H

( j)



then

← C

− { j};7

EXPAND(S,{i},C

,τ);8

procedure EXPAND(S,I,C,τ)

/* I,C ⊆ {1, . ..,n} */

← C;9

foreach i ∈ C do10

← C

\ {i};11

if R

(I ∪ {i}) ≥ τ then12

output I ∪ {i};13

EXPAND(S,I ∪ {i},C

,τ);

Algorithm 1: FINDCM.

set of the above characters is regarded as Σ. Also we

insert a gap symbol ‘-’ at the position where an amino

acid is deleted. Furthermore, since a character ’X’

is inserted if an amino acid is unknown in the amino

acid sequences provided from NCBI

, we replace ‘X’

at a position as the most frequent amino acid at the

position in all the amino acid sequences.

3.1 Running Time

In this subsection, we investigate the running time of

the algorithm FINDCM. Here, the computer environ-

ment is that OS is WSL2 over Windows 10, CPU is

Intel Core i7-7700 3.6GHz and RAM is 16GB.

Table 2 and Table 3 illustrate the running time

of the algorithm FINDCM for amino acid sequences

of the Delta variant and the Omicron variant, respec-

tively.

Tables 2 and 3 show that, for 0.80 ≤ τ ≤ 0.90, the

computation time of the Delta variant is smaller than

that of the Omicron variant for ρ = 0.10 and 0.05, the

computation time of the Delta variant is much larger

than that of the Omicron variant for ρ = 0.01. Note

that the number of sequences for the Delta variant is

834,870 and that for the Omicron variant is 393,655,

so the former is more than twice the latter. Hence, by

comparing the computation time for FINDCM with

the number of the sequences, the number of candi-

dates, not pruned by τ and ρ in FINDCM, of the corre-

lated mutations for the Omicron variant is much larger

than that for the Delta variant for small ρ.

ICPRAM 2023 - 12th International Conference on Pattern Recognition Applications and Methods

346

Table 2: The running time (sec.) of the algorithm FINDCM

for amino acid sequences of the Delta variant.

τ ρ time (sec.)

0.90 0.10 56

0.05 74

0.01 1,366

0.85 0.10 89

0.05 114

0.01 2,089

0.80 0.10 144

0.05 173

0.01 2,817

Table 3: The running time (sec.) of the algorithm FINDCM

for amino acid sequences of the Omicron variant.

τ ρ time (sec.)

0.90 0.10 2

0.05 9

0.01 11,840

0.85 0.10 3

0.05 11

0.01 80,522

0.80 0.10 4

0.05 16

0.01 427,746

τ ρ time (sec.)

0.75 0.10 5

0.70 5

0.65 5

0.60 6

0.55 6

0.50 7

0.45 21

0.40 30

0.35 40

0.30 78

0.25 127

0.20 242

3.2 Correlated Mutation for Delta

Variant

In the following subsections, we discuss the corre-

lated mutations found by the algorithm FINDCM.

Here, we denote the correlated mutations by the set

of the following form:

⟨structural protein⟩position.

In this subsection, we investigate the found cor-

related mutations for the Delta variant. When we ﬁx

ρ = 0.01, Table 4 illustrates the found correlated mu-

tations for the Delta variant obtained by varying τ

as 0.90, 0,85 and 0.80. Here, “id” means the index

“CMi” of the i-th correlated mutations. Also, we de-

note the positions in the spike protein substitutions for

SARS-CoV-2 Delta and Omicron variants in Table 1

in Section 1 by bold faces. Furthermore, we denote

the newly added positions in a correlated mutation as

underlined.

Table 4 shows that, whereas the correlated muta-

tions of CM1 and CM4 do not change even if τ is

varied from 0.90 to 0.80, the correlated mutations of

CM2 and CM3 are added to new positions in corre-

lated mutations. Also, all of the correlated mutations

Table 4: The correlated mutations for the Delta variant ob-

tained by varying τ as 0.90, 0,85 and 0.80 under ρ = 0.01.

τ id correlated mutation

0.90 CM1 ⟨S⟩570, ⟨S⟩716, ⟨S⟩982, ⟨S⟩1118, ⟨N⟩3,

⟨N⟩235

CM2 ⟨S⟩452, ⟨S⟩478, ⟨M⟩82

CM3 ⟨S⟩190, ⟨N⟩80

CM4 ⟨S⟩501, ⟨N⟩204

0.85 CM1 ⟨S⟩570, ⟨S⟩716, ⟨S⟩982, ⟨S⟩1118, ⟨N⟩3,

⟨N⟩235

CM2 ⟨S⟩19, ⟨S⟩452, ⟨S⟩478, ⟨M⟩82, ⟨N⟩377

CM3 ⟨S⟩20, ⟨S⟩190, ⟨S⟩655, ⟨N⟩80

CM4 ⟨S⟩501, ⟨N⟩204

0.80 CM1 ⟨S⟩570, ⟨S⟩716, ⟨S⟩982, ⟨S⟩1118, ⟨N⟩3,

⟨N⟩235

CM2 ⟨S⟩19, ⟨S⟩452, ⟨S⟩478, ⟨S⟩681, ⟨S⟩950,

⟨M⟩82, ⟨N⟩63, ⟨N⟩203, ⟨N⟩377

CM3 ⟨S⟩20, ⟨S⟩26, ⟨S⟩190, ⟨S⟩655, ⟨S⟩1027,

⟨S⟩1176, ⟨N⟩80

CM4 ⟨S⟩501, ⟨N⟩204

CM5 ⟨S⟩289, ⟨N⟩18

in Table 4 contain the positions of S and the other

structural proteins.

In particular, the correlated mutation CM2 at τ =

0.80 contains ﬁve positions for spike protein substitu-

tions that are just positions in S. Note that the ﬁve po-

sitions are known to mainly characterize SARS-CoV-

2 Delta variant (Chen et al., 2022).

3.3 Correlated Mutation for Omicron

Variant Under ρ = 0.01

In this and next subsections, we investigate the corre-

lated mutations for Omicron variant.

First, when we ﬁx ρ = 0.01, Table 5 illustrates the

found correlated mutations for the Omicron variant

obtained by varying τ as 0.90, 0,85 and 0.80. Here,

“+” means that the correlated mutation is obtained by

adding the upper correlated mutation to the presented

positions.

Table 5: The correlated mutations for the Omicron variant

obtained by varying τ as 0.90, 0,85 and 0.80 under ρ = 0.01.

τ correlated mutation

0.90 ⟨S⟩109, ⟨S⟩110, ⟨S⟩114, ⟨S⟩115, ⟨S⟩116, ⟨S⟩117,

⟨S⟩119, ⟨S⟩121, ⟨S⟩122, ⟨S⟩125, ⟨S⟩126, ⟨S⟩129,

⟨S⟩131, ⟨S⟩136, ⟨S⟩139, ⟨S⟩140, ⟨S⟩149

0.85 + ⟨S⟩113, ⟨S⟩120, ⟨S⟩122, ⟨S⟩124, ⟨S⟩127, ⟨S⟩137,

⟨S⟩144, ⟨S⟩150

0.80 + ⟨S⟩130, ⟨S⟩134, ⟨S⟩141, ⟨S⟩154

Correlated Mutations of Positions Among Structural Proteins in Delta and Omicron Variants for SARS-CoV-2 Amino Acid Sequences

347

Table 5 shows that, under ρ = 0.01 and 0.80 ≤

τ ≤ 0.90, we can ﬁnd one correlated mutation, which

is concerned with just S with positions from 109 to

154. Also the position of 144 is concerned with spike

protein substation in Table 1.

3.4 Correlated Mutation for Omicron

Variant Under ρ = 0.10

Next, when we ﬁx ρ = 0.10, Table 6 illustrates the

found correlated mutations for the Omicron variant

obtained by varying τ from 0.70 to 0.20 decreasing

by 0.05.

Table 6: The correlated mutations for the Omicron variant

obtained by varying τ from 0.70 to 0.20 decreasing by 0.05

under ρ = 0.10.

τ id correlated mutation

0.70 CM1 ⟨S⟩289, ⟨N⟩18

0.65–0.55 CM1 ⟨S⟩289, ⟨N⟩18

CM2 ⟨S⟩222, ⟨N⟩215

0.50 CM1 ⟨S⟩289, ⟨N⟩18

CM2 ⟨S⟩112, ⟨S⟩222, ⟨N⟩215

0.45 CM1 ⟨S⟩5, ⟨S⟩289, ⟨S⟩809, ⟨S⟩1104,

⟨S⟩1264, ⟨N⟩9, ⟨N⟩18, ⟨N⟩63

CM2 ⟨S⟩112, ⟨S⟩222, ⟨N⟩215

CM3 ⟨S⟩95, ⟨S⟩142

0.40–0.35 CM4 ⟨S⟩5, ⟨S⟩112, ⟨S⟩222, ⟨S⟩289, ⟨S⟩809,

⟨S⟩1104, ⟨S⟩1264, ⟨N⟩9, ⟨N⟩18,

⟨N⟩63, ⟨N⟩215

CM3 ⟨S⟩95, ⟨S⟩142

0.30–0.20 CM5 ⟨S⟩5, ⟨S⟩95, ⟨S⟩112, ⟨S⟩142, ⟨S⟩222,

⟨S⟩289, ⟨S⟩809, ⟨S⟩1104, ⟨S⟩1264,

⟨N⟩9, ⟨N⟩18, ⟨N⟩63, ⟨N⟩215

Table 6 shows that all of the correlated mutations

contain the positions of S and the other structural pro-

teins. Also, the correlated mutation CM3 contains the

positions for spike protein substitutions.

Also Table 6 shows that the correlated mutation

CM4 is the combination of CM1 and CM2, and the

correlated mutation CM5 is the combination of CM3

and CM4. Hence, we can regard that the correlated

mutation CM5 is the convergence of other found cor-

related mutations.

On the other hand, the number of positions in

spike protein substitutions occurring in the correlated

mutations is just two, which is small. Then, it is a fu-

ture work to ﬁnd the correlated mutations containing

more positions in spike protein substitutions.

4 CONCLUSION

In this paper, we have found the correlated mutations

of positions among structural proteins in amino acid

sequences of SARS-CoV-2 Delta and Omicron vari-

ants by using the algorithm FINDCM designed by

(Shimada et al., 2012). Then, we have obtained the

correlated mutations containing the positions among

several structural proteins and containing the posi-

tions occurring in the spike protein substitutions in

SARS-CoV-2 Delta and Omicron variant.

In particular, we have found the correlated muta-

tion CM5 in Table 6 as the convergence of several

correlated mutations containing the positions in the

spike protein substitutions. On the other hand, it is a

future work to investigate the positions except a spike

protein of CM2 at τ = 0.80 in Table 4 and CM5 in

Table 6 in the genomic viewpoints.

The algorithm FINDCM is based on the set enu-

meration algorithm (Rymon, 1992). Then, it is a fu-

ture work to design the algorithm of ﬁnding correlated

mutations based on another enumeration algorithm,

with introducing another thresholds like τ and ρ.

Whereas we have found the correlated mutations

concerned with the positions in the spike protein sub-

stitutions, the number of them is small, in particular,

for the Delta variant. Also, whereas the algorithm

FINDCM ﬁnds all of the correlated mutations under

given τ and ρ, it is necessary to ﬁnd the correlated

mutations concerned with the positions in the spike

protein substitutions directly and efﬁciently. Hence,

it is a future work to design a new algorithm of ﬁnd-

ing the correlated mutations containing given several

positions like as the positions in spike protein sub-

stitutions, which is possible to be more efﬁcient than

FINDCM.

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