Visual and Quantitative Analyses of Virus Genomic Sequences using a
Metric-based Algorithm
ALEXANDRA BELINSKY
Mach-3dP Inc.
Burlington, ON, CANADA
GUENNADI A. KOUZAEV
Norwegian University of Science and Technology - NTNU
Trondheim, NORWAY
Abstract: - This work aims to study the virus RNAs using a novel accelerated algorithm to explore any-
length repetitive genomic fragments in sequences using Hamming distance between the binary-
expressed characters of an RNA and a query pattern. Primary attention is paid to the building and
analyzing 1-D distributions (walks) of atg-patterns - codon-starting triplets in genomes. These triplets
compose a distributed set called a word scheme of RNA. A complete genome map is built by plotting
the mentioned atg-walks, trajectories of separate (a-, c-, g-, and t-symbols) nucleotides, and the lines
designating the genomic words. The said map can be additionally equipped by gene’s designations
making this tool pertinent for multi-scale genomic analyses. The visual examination of atg-walks is
followed by calculating statistical parameters of genomic sequences, including estimating walk-
geometry deviation of RNAs and fractal properties of word-length distributions. This approach is
applied to the SARS CoV-2, MERS CoV, Dengue, and Ebola viruses, whose complete genomic
sequences are taken from GenBank and GISAID. The relative stability of these walks for SARS CoV-
2 and MERS CoV viruses was found, unlike the Dengue and Ebola distributions that showed an
increased deviation of their geometrical and fractal characteristics. The developed approach can be
useful in further studying mutations of viruses and building their phylogenic trees.
Key-Words: - Biological control systems, Modeling of Biological Systems, RNA,
Hamming-distance metric measure, quantitative RNA walks, SARS Cov-2 virus, MERS CoV virus,
Dengue virus, Ebola virus
Received: April 19, 2022. Revised: November 5, 2022. Accepted: November 24, 2022. Published: December 31, 2022.
1 Introduction
A virus is a tiny semi-live unit carrying genetic
material (RNA or DNA – double-helix RNA
structure) in a protein capsid covered by a lipid
coat. The virus penetrates the cell wall and
urges this bio-machine to ‘manufacture’ more
viruses.
Some viruses are RNA-based and transfer
genetic information by long chains of four
organic acids, namely, Adenine (a), Cytosine
(c), Guanine (g), and Uracil (u) [1]. DNA-based
viruses and double-stranded genetic polymers
carry the information by four nucleotides, but
one of them is Thymine (t) instead of Uracil. In
genomic databases, even the single-stranded
viral RNAs are registered as complementary
chains where Thymine substitutes Uracil due to
some instrumental specifics [2] that do not
hinder the mathematical aspects of the virus
theories. These complimentary RNAs will be
used further for numerical modeling in our
paper.
A complete RNA is a chain of codons
(exons) used to transfer genetic information and
introns. Unfortunately, the latter’s role is not
well known [3]. Sequencing of RNA or DNA is
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Volume 21, 2022
searching and identifying nucleotides by
instrumental means. Codons in RNAs start with
an ‘aug’ combination of nucleotides and end
with one of the following three combinations:
‘uaa’, ‘uag’, or ‘uga’. Additionally, ‘aug’ may
play a coding role in genomes. Some DNA
strands consist of billions of nucleotides, so
mathematical methods are widely used in
genomics [4]. For instance, the RNA symbols
are substituted by number values, and this
process is called DNA/RNA mapping [5]-[8].
For example, in Ref. [5], eleven methods of
numerical representation of genomic sequences
are listed and analyzed to conclude that each is
preferable in a particular application, and no
universal mapping algorithm is equally
advantageous for all genomic studies.
Different retrieval algorithms can be
applied to genomic sequences, including signal-
processing means [7],[9]-[14]. The numerical
RNAs can be shown graphically for qualitative
analyses. For instance, each nucleotide is
represented by a unit vector in a 4-dimensional
(4-D) space, and an imaginary walker moves
along an RNA sequence, making a trajectory in
this space that can be described by fractional
order differential or integrodifferential
equations [15],[16].
The nucleotides are combined in a certain
way to avoid apparent difficulties with plotting
walks in multidimensional spaces [17],[18]. For
instance, each nucleotide is associated with one
of four unit vectors in 2-D space, which
projections on the plane axes can take positive
(+1) or negative (–1) values [19],[20]. A
trajectory is built moving along the consecutive
number of a nucleotide in the studied genomic
sequence.
In general, DNA walks allow the detection
of codons and introns, discover hidden RNA
periodicity [12]-[14], and calculate
phylogenetic distances between genomic
sequences [21], among others. Some additional
results and reviews on DNA imaging can be
found, for instance, in Refs. [17],[22]-[24],
where the necessity to use specified walks for
each class of genomic problems is shown.
Genomic walk analysis can be followed by
calculating fractal properties of distributions of
nucleotides [25]-[34]. Fractals are self-similar
or scale-invariant objects. It means small ‘sub-
chain’ geometry is repeated on larger geometry
scales, although it is randomly distorted.
A biopolymer chain in a solution is bent
fractally [28]. This fractality influences the
chemical reaction rate, diffusion, and surface
absorption of long-chain and globular
molecules, among others [27],[28],[35]-[38].
Because polar solvents have frequency-
dependent properties, they are adjusted by
microwaves that influence the polymer fractal
dimension. Thus, some bioreactions can be
controlled by a weak high-gradient microwave
field [39],[40].
Although many achievements are known in
the numerical mapping of RNAs, some
questions have not been resolved. For instance,
the known genomic walks are designed to track
single nucleotides or their pairs, leading to
crowdy trajectories and overloaded plots that
are challenging to analyze visually in one plot
[5]-[8].
Meanwhile, the complete RNAs of viruses
are composed chiefly of codons, and one
repetitive pattern therein is their starting atg-
triplet. We assign to these triplets a
mathematical construction - a viral RNA
scheme.
Our proposed pattern search algorithm
calculates the triplet distributions along an RNA
sequence. Additionally, the same algorithm can
make the walks of each of the four nucleotides.
These trajectories are found not twisted firmly
in comparison to curves from Refs. [5],[8], for
instance, and they are easier to be analyzed
visually. These graphs can be equipped with
marks pointing to genes and hyperlinks with the
gene names, making these figures interactive
means to analyze the genomes.
Some results of creating such a tool and
applications to genomes of several viruses are
given here. The source codes and visualizations
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can be used for research and practical
applications. One of them is the studies of the
stability of the mentioned atg-schemes towards
mutations, variation of codon fillings, and the
fractality of atg-distributions, among others.
Section 2 considers our proposed calculation
algorithms and plotting techniques in detail
with application examples. The results of
applying these techniques to the SARS CoV-2,
MERS CoV, Dengue, and Ebola viruses are in
Section 3. They are further discussed in detail in
Section 4, and the conclusions are in Section 5.
In Appendix 1, all necessary data for the
analyzed RNAs taken from GenBank and
GISAID are tabular, including the parameters
calculated in this contribution.
2 Materials and Methods
2.1 Materials and Data Availability
In this paper, only complete arbitrary-chosen
genomic sequences without missed symbols are
studied taken from GenBank [41] and GISAID
[42]. Among them are 36 SARS CoV-2
genomic sequences from GISAID and one from
GenBank, 20 genomic sequences for the MERS
coronavirus (GenBank), 25 for the Dengue
virus from GenBank, and 15 Ebola virus
genomic sequences (GenBank). Data from the
GISAID are available after registration. All
names of genomic sequences are given in figure
legends and Tables 1–4.
2.2 Methods and Application Examples
2.2.1 Metric-based atg-triplet Algorithm
As has been stated above, for both DNA and
RNA descriptions by characters, their alphabet
consists of four nucleotides marked by a-, c-, g-
, and t-symbols. These designations are used to
study RNA or DNA if their physicochemical
properties are outside the research scope.
In many cases, RNA/DNA have repeated
patterns of nucleotide sequences, and these
regions are better conserved in mutations [9],
[43],[44]. As a rule, pattern discovery relates to
nondeterministic polynomial time problems
(NP-problems), i.e., solution time increases
exponentially with the sequence length.
A typical algorithm compares a query
character pattern with a length of nucleotides
with the following one-symbol shift of query
along a chain. In our code, we use these
techniques, as well. Usually, the search
algorithms working with characters with no
assigned numerical values are slower by 1.43-
2.37 times than those processing the binary
variables, as shown in Refs. [45],[46].
Moreover, even the computer arithmetic logic
units can be re-designed to fulfill the frequently
repeated patterns of binary operations to
accelerate them [47]. Then, in our code, all
RNA’s characters are transformed into binaries
before all operations using a Matlab operator
dec2bin(character) [48].
In computers, for instance, the UTF-8
format allows encoding all 1,112,064 valid
character code points, and it is widely used for
the World Wide Web [49]. The first 128
characters (US-ASCII) require only one byte
(eight binary numbers) in this format. If binary
units initially represent the sequences, then
calculating the DNA sequence’s numerical
properties in this form reduces the computation
time.
Because the binary sequences now write the
DNA/RNA chains, they can be characterized
quantitatively using a suitable technique. One
calculates a metric distance between the binary-
represented symbols and a query (base)
‘moving’ along a chain.
Many metric types are used in codes and big
data [50]-[55]. The advantage of using metric
estimates is that they can be applied in cluster
analysis for similar grouping nucleotide or
protein distribution patterns. For instance, it can
help classify the virus RNAs [56],[57]. Notably,
this distance can be the Hamming one [50] used
further in this paper.
Consider a flow chart of our code on
exploring patterns of arbitrary length
n
(Figure
1). It starts with importing sequence data
A
of
the length
N
from any genomic database in
FASTA format [41],[42] and defining a query
pattern
B
of any size
n
. From both files, the
empty spots should be removed [58].
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Fig. 1: Algorithm flowchart.
In the second step, the data files are
transformed into binary strings, which are used
to calculate Hamming distance between each
binary symbol of
A
and
B
. This distance is a
metric for comparing two binary values, and it
is the number of bit positions in which the two
bits are different.
To calculate Hamming distance
H
d
between two strings
A
and
B
, the
XOR
operation
AB
is used, and the total number
of ‘1’s in the resultant string
C
is counted. The
distance value is zero if the compared binary-
represented symbols are the same. Only
n
characters are compared on each count; then,
the query is moved to one symbol, and
calculations are repeated.
Then, a string
C
of numerical estimates of
the length
M n N
is a product of step 3 of
this code. Not all registered genomic sequences
are divisible by
n
. In this case, a needed
number of characters
a
is added to the end of
the sequence
A
, or the Hamming metric
operation can be fulfilled using Levenstein’s
distance formula from Ref. [53], workable for
compared sub-strings of arbitrary length.
The following two parts of our algorithm are
calculating numbered
i
y
query positions in a
sequence
i
x
according to the Hamming-
distance data. All zeros in the string
C
are
initially obtained (step 4, Figure 1). Then, only
n
neighboring zeros corresponding to a query
are selected (step 5, Figure 1), and this query is
numbered in a sequential manner starting with
the first one found in RNA. The positions
i
x
of
these numbered queries
i
y
in a complete RNA
sequence are calculated analytically (step 6,
Figure 1).
Let us take the coordinate of the first symbol
in a numbered found query, then a set of points
can be built along a studied sequence. These
points, being connected, make a curve called a
query walk.
In this paper, the codon start-up atg-triplet is
used below as a query (pattern
B
). We define
the positions
i
x
of the first symbols of the
sequentially numbered atg-triplets in an RNA
A
, and the atg-walks are plotted.
Additionally, we calculate the word length
, 1 1
atg
i i i i
l x x


. In our algorithm, a ‘word’ is a
nucleotide sequence starting with an atg-triplet
and all symbols up to the next atg-one (Figure
1, right side).
The proposed algorithm was realized in the
Matlab environment [48], and it is a few-ten-
line code. The following Matlab library
functions were used:
1.
2 (character variable)dec bin
– to
transform a character variable into a
binary one
2.
2 , , ‘ ’ptisd a b hamming
– to calculate
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the Hamming distance value between
two binary values
a
and
b
3.
stringzeros
– calculation of numbers
of zero-values in a string
4.
,plot y x
– plot function
yx
The developed algorithm was applied to
many available virus complete genomes.
Because the atg-triplets start the codons in most
cases, the main attention was paid to building
the distributions of these repeated patterns
called the RNA word schemes. The atg-
positions were compared with the found ones
from the given genomes to verify these
calculations.
2.2.2 Visualization Techniques
2.2.2.1 One-dimensional atg-walks
The viral RNAs, consisting of thousands of
nucleotides, are challenging to analyze, and
many visualizing methods are used. Among
these techniques is the plotting the DNA walks
projected on the spaces of appropriated
dimensions considered above (see
Introduction). Some contributions are full of
symbolic designations of nucleotides and
diagrams showing the positions of genes in the
complete DNA sequences. The preference for a
visualization method is dictated by the
specificity of applications, although there is a
need for a universal graphical tool.
This paper found that atg-walks could be
plotted by coordinates of the numbered atg-
triplets in a complete RNA sequence. Like the
known DNA walks, these dots can be
considered the points of a trajectory named here
as atg-walk (Figure 2A). A diagram showing
the positions of the symbol
a
in defined atg-
triplets is given in Figure 2B.
Similar to single-symbol DNA walks, the
atg-trajectories have fractal properties. Their
type was defined by analyzing distributions of
the coordinates of triplets shown by vertical
lines along an RNA sequence (Figure 2B).
These atg-distributions have repeating motifs
on different geometry scale levels, i.e., they
have fractal properties.
Fig. 2: Positions of atg-triplets along the genome
sequence of SARS-CoV-2 virus MN988668.1 (row 1,
Table 1, Appendix 1) given for the first 5 000 nucleotides
provided by points (A) and vertical lines in diagram (B).
Below, our initial assumption about the
fractality of atg-distributions will be confirmed:
we calculated the fractal dimensions of
complete genomes of several tens of virus
sequences. Presumably, the atg-triplets are
distributed along with the RNA sequences of
studied viruses according to the random Cantor
multifractal set rule [59].
2.2.2.2. Multi-scale Mapping of RNA Sequences
It is necessary to see full-scale virus RNA maps
and analyze all types of mutations. Previously,
the most attention was paid to mapping atg-
triplets, thinking they constructed a scheme of
RNA, a relatively stable structure. Besides the
structural mutations changing the atg-
distributions, the nucleotides vary their
positions inside codons. Our algorithm
considers even a single symbol as a pattern,
allowing the calculation of distribution curves
for each nucleotide similar to atg-ones. These
curves can be considered the first level of
spatial detailing of RNAs. The words in our
definitions (see Figure 1) compose the second
level. Words form a gene responsible for
synthesizing several proteins, and the genes
belong to the third level of spatial detailing of
RNAs.
A combined plotting of elements of the
hierarchical RNAs organization will be helpful
in the visual analyses of genomes. One of the
ways is shown in Figure 3.
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Fig. 3: Two-scale study results of a SARS-CoV-2 virus
MN988668.1 (row 1, Table 1, Appendix 1). In the inlet,
these symbols are pointed inside the words given for the
first 300 nucleotides.
Here, positions of a-symbols of atg-triplets
in a sequence are given by vertical lines (second
level of detailing). Words take spaces between
these vertical lines (see Figure 1). They are
filled by numbered nucleotides (points of a
different color), which are the first level of
RNA detailing. It allows distinguishing
nucleotides even at the beginning of coordinates
where visual clutter is seen (inlet in Figure 3).
The next level of hierarchical RNAs
organization is with genes. For instance, in
GenBank [41], the list of symbols of a genomic
sequence in FASTA format is followed by a
diagram where the genes are given by
horizontal bars with the gene’s literal
designations.
In our case, this diagram can be attached to
a two-scale plot considered above. Another
solution is to equip our figures with gene
hyperlinks, an interactive means highlighting
whose genes a nucleotide or codon belongs to,
shown by a pointer.
Thus, the developed pattern search algorithm
based on the Hamming distance applied to
binary representations of nucleotide symbols
allows building combined plotting of
hierarchical organization of the RNAs of
viruses. It can also be applied to the analyses of
more complex protein structures. Different from
many genomic walks, it produces spatially
simple curves that can be analyzed visually and
quantitatively.
2.2.3 Calculation Techniques for Fractal
Properties of atg-Distributions
In many previous studies, the fractality of
DNA/RNA sequences has been studied
[6],[20]-[26],[29]-[38]. The motifs of small-size
genetic patterns are repeated on large-scale
levels. Thus, the nucleotide distribution along a
genome is not entirely random due to this long-
range fractal correlation, as is mentioned in
many papers.
The large-size genomic data are often
patterned, and each pattern can have its fractal
dimension, i.e., the sequences can be
multifractals [33]. This effect is typical in
genomics, but it is also common in the theory of
nonlinear dynamical systems, signal processing,
and brain tissue morphology, among others
[60]-[68].
Discovering the fractality of genomic
sequences is preceded by their numerical
representation, for instance, by walks of
different types [6]-[8],[20],[22],[29],[33]. Then,
each step value of a chosen walk is considered a
sample of a continuous function, and the
methods of signal processing theory are applied
[12],[13].
The measure of self-similarity is its fractal
dimension
F
d
which can be calculated using
different methods. In our case, the fractal
dimension calculations can be applied directly
to the distribution of atg-consecutive numbers
i
y
(Figure 2a), but it gives this value close to 1
for the analyzed RNAs, i.e., the dependence
()
ii
yx
is close to the linear one. Then, these
calculations are not effective in analyses due to
their weak sensitivity. Instead, the word-length
distributions along RNAs sequences
,1
atg
ii
l
(see
sub-section 2.2.1 and Figure 1, right side) are
used.
A particular distribution of the word lengths
is shown in Figure 4 by bars whose heights
equal the word lengths. Then, the algorithms,
usually applied to the sampled signals, can be
used to compute the statistical properties of
these word-length distributions.
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Fig. 4: Word-length distribution in a SARS Cov-2 virus
MN988668.1 sequence (row 1, Table 1, Appendix 1).
In this paper, the fractal dimension of word
length distributions was calculated using the
software package FracLab 2.2 [67]. This code
provides results with reasonable accuracy if the
default parameters of FracLab are used.
Although many researchers tested this code, it
is again verified to calculate the Weierstrass
function, which is synthesized according to a
given fractal dimension value [68].
In a strong sense, the fractal dimension was
defined for the infinite sequences. In our case,
the studied RNAs have only 268-730 atg-
triplets depending on the virus. Then, the fractal
dimension values were estimated
approximately.
3 Results
3.1. Study of atg-walks of SARS CoV-2
Genome Sequences
In this research, essential attention was paid to
studying SARS CoV-2 complete RNA genome
sequences. A recent comprehensive review of
the genomics of this virus can be found in Refs.
[69],[70], for instance.
The data used here and throughout this
whole paper are from two genetic databases:
GenBank [41] and GISAID [42]. A part of the
studied genome sequences for this and other
viruses is provided in Appendix 1.
Here, the main unit, called a ‘word’, is a
nucleotide sequence starting with ‘atg’ and the
symbols up to the next triplet (Figure 1). The
number of atg-s was calculated by our code and
verified by a Matlab function
, ' 'count A atg
.
These results are shown in the third column of
Tables 1–4 (See Appendix 1). The Matlab
functions
word
median L
and
word
rms L
calculated the median and root-mean-square
(R.M.S) values of each sequence’s word-length
,1
atg
ii
l
distribution correspondingly. The results
are in columns 4 and 5 of the mentioned tables.
Consider applying the developed approach
to the complete genome of a Wuhan RNA
sample MN988668.1 (GenBank) as an example.
It consists of 29881 nucleotides and 725 atg-
triplets (See row 1, Table 1, Appendix 1).
Figure 2 shows the distribution by points of atg-
triplets for the first 5000 nucleotides of this
complete genome. In Figure 3, the entire study
of this virus is shown. Word-length distribution
of this sample is given in Figure 4.
Figure 5 illustrates the distribution (in lines)
of atg-triplets along with complete genome
sequences for thirty-seven SARS CoV-2
arbitrary-chosen virus samples, including Delta,
Omicron, and a bat-corona sample (see Table 1,
Appendix 1).
There were relatively compact localizations
of the triplet curves despite the viruses being of
different clades and lines. For instance, the
relative difference
1,21 1 21
( 29455) 100% 2
i
y x y y y
of
these curves 1-37 (Figure 5) is estimated at
29455
i
x
around only 1.6%. This confirms
the conclusions of many specialists [71] that no
new recombined strains have appeared up to
this moment, despite many mutations found to
date (2023), including the Omicron lineage.
Two inlets show the beginning and tails of
these curves to illustrate details. Although, in
general, these trajectories are woven firmly, the
tails are between the bat’s SARS-CoV-2 light-
blue curve (hCoV-
19/bat/Cambodia/RShSTT182/2010, row 6,
Table 1, Appendix 1) and the black trajectory
obtained for a sequence from Brazil (hCoV-
19/Brazil/RS-00674HM_LMM52649/2020, row
14, Table 1, Appendix 1).
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Fig. 5: Distributions of atg-triplets of 31 SARS Cov-2
complete RNA sequences (Table 1, Appendix 1). The
inlets show these atg-distributions at the beginning and
end of genome sequences.
A detailed study of viruses from Table 1,
Appendix 1 shows that each considered here
sequence has individual atg-distribution. It
means that most mutations are combined with
the joint variations of word content, word
length, and the number of these words. Other
mutations with only word content variation may
exist. However, the atg-walks cannot see them,
and the single-symbol distributions considered
below will help us to detect these modifications
of viruses (See Section 2.2.2.2).
Figure 6 shows a detailed comparison of
samples of five viruses causing increased
trouble for specialists with the one from
Wuhan, China. The inlets offer the details of
these curves at their beginning and end. The
tails of the three curves are closed between the
Wuhan and Brazil trajectories. Although the
difference between these curves is not
significant, the mutations may have
complicated consequences in the rate of
contagiousness of viruses.
Different techniques for numerical
comparison of sequences are known from data
analytics, including, for instance, calculation of
correlation coefficients of unstructured data
sequences, data distance values, and clustering
of data, among others [10],[56],[57].
Researching RNA sequences, we suppose that
the error of nucleotide detection is essentially
less than one percent; otherwise, the results of
comparisons would be instrumentally noisy
Fig. 6: Detailed distributions of atg-triplets for five
trouble-making SARS Cov-2 complete RNA sequences
(rows 1,8,3,14,19, Table 1, Appendix 1). The inlets show
these atg-distributions at the beginning and end of
genome sequences.
We use a simplified algorithm for
quantitative comparing atg-distributions of
different virus samples. Each numbered atg-
triplet
i
y
has its coordinate along a sequence
i
x
. Thanks to mutations, the length of some
coding words varied together with the
coordinate
i
x
of a triplet.
In our case, we calculated the difference
(deviation) between the coordinates
i
x
of atg-
triplets of the same numbers
i
y
in the
compared sequences. This operation was
fulfilled only for the sequences of equal atg-
triplets; otherwise, excessive coding words are
neglected in comparisons. Of course, such a
technique for the comparison of geometrical
data has its disadvantages. Therefore, if a
compared sequence has several atg-triplets
fewer than the number of atg-ones in a
reference sequence, the atgs of reference RNA
are excluded from comparisons. Still, it allows
for obtaining some information on mutations of
viruses in a straightforward and resultative way
that will be seen below.
Our approach supposes choosing a reference
nucleotide sequence to compare the genomic
virus data of other samples, and it is a complete
genomic sequence MN988668.1 from GenBank
(row 1, Table 1). Several virus samples from
GenBank and GISAID have been studied in this
way [44], and some results of comparisons are
given in Figure 7.
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Fig. 7: Deviation of atg-coordinates in RNAs of four
SARS CoV-2 viruses relative to the reference RNA
MN988668.1 (see Table 1, Appendix 1).
The ordinary axis
x
in these plots shows
the deviation of coordinates
i
x
of atg-triplets
from the atg-coordinates of the reference
sequence. As a rule, due to the different number
of noncoding nucleotides at the beginning of
complete RNA sequences, the curves in Figure
7 have constant biasing along the
x
axis.
The straight parts of these curves mean that
the atg positions of a compared sequence are
not perturbed regarding the corresponding
coordinates in the reference RNA. It means that
there are no mutations, or they are only with the
variation of coding words without affecting
their lengths if these mutations have a place.
In some studied samples (here and in Refs.
[44],[72]), perturbations are near the end of the
orf1ab gene, as is seen using a graphical tool of
GenBank [41]. Perturbations were detected by
calculating the
i
x
coordinates according to
known
i
y
. The atg-perturbations could
generally occur in any RNA’s part, considering
the random nature of mutations (Figure 7C, D).
Relative deviation
i
xN
did not exceed 1–2%
for compared viruses. Although this deviation is
mathematically tiny, it may have severe
consequences in the biological sense.
Our study shows that these difference curves
(Figure 7) are individual for the studied
samples. Although mutations without affecting
the atg-distributions are possible, this
individuality, theoretically, may be lost.
There are repeating motifs of comparison
curves (Figure 7 and Refs. [44], [72]). The
origin of this is unknown, but it was not
coupled with the lineages of viruses and their
clades. Other viruses can be studied in a similar
manner.
3.2. Study of atg-walks of Complete Genome
Sequences of the Middle East Respiratory
Syndrome-related Coronavirus
Middle East Respiratory Syndrome-related
(MERS) is a viral illness. The virus’ origin is
unknown, but it initially spread through camels
and was first registered in Saudi Arabia
[73],[74]. Most people infected with the MERS
CoV virus developed a severe respiratory
disease, which resulted in multiple human
deaths.
Our simulation of atg-distributions of this
virus shows the compactness of the calculated
curves (Figure 8, and Table 2, Appendix 1), like
the SARS CoV-2 characteristics. It follows that
both viruses demonstrate relatively stable
features towards the strong mutations connected
with the recombination of the virus’s parts. For
instance, the relative difference
1,10 29982
i
yx
of these curves is estimated
at around 1% only. On average, the MERS
RNAs have fewer number of atg-triplets, and
longer nucleotide words than the SARS CoV-2
studied sequences.
In general, according to our more than three-
year observation, the two studied coronaviruses
(MERS CoV and SARS CoV-2) demonstrate
relatively strong stability of their atg-
distributions towards severe mutations, leading
to the variation of codon positions, word
lengths, and the number of words. It follows the
conclusions of many scientists working in
virology and virus genomics [71].
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Fig. 8: Distributions of atg-triplets of 20 samples of the
MERS CoV complete RNA sequences (Table 2,
Appendix 1). Inlet shows the atg-distributions at the end
of genome sequences. The inlets offer these distributions
at the beginning and end of genome sequences.
3.3. Dengue Virus Study
The Dengue virus is spread through mosquito
bites. For instance, a comprehensive review of
the genomics of this virus can be found in Refs.
[75],[76].
Unlike the coronaviruses, the Dengue virus
(Table 3, Appendix 1) tends to form separate
families, i.e., it is less stable than SARS-CoV-2
and MERS viruses. It has five genotypes
(DENV 1–5) and around 47 strains.
Unfortunately, the genomic data for this virus
have been published less compared to
coronaviruses. Some of them for which the
complete genome data are available from
GenBank have been studied below.
Figure 9A (rows 1-5, Table 3, Appendix 1)
shows the atg-distributions of five sequences of
Dengue virus-1 found in China. A rather large
dispersion of these sequences is seen from these
graphs.
Figure 9B gives the atg-distributions of five
complete sequences of the Dengue virus-2.
These distributions are more compactly
localized, although their origin is from different
parts of the world. In general, the observed
Dengue virus-2 samples have an increased
number of shorter words compared to the
sequences of Dengue virus-1.
Fig. 9: Distributions of atg-triplets of complete RNA
sequences of the Dengue virus-1 - (A) and Dengue virus-
2 - (B). See rows 1-5 and 6-10, correspondingly from
Table 3, Appendix 1.
Figure 10A shows five data sets for different
strains of Dengue virus-3 registered in many
countries. They have about the same number of
nucleotides, and comparable averaged lengths
of words.
Figure 10B gives three atg-distributions of a
Gabon-strain [76] of Dengue virus-3. It is
supposed that this strain mutated from the
earlier registered Dengue virus lines (Figure
10A). However, they are different in the length
of complete genome sequences and their
statistical characteristics, which will be
considered in Section 3.5 below.
Fig. 10: Distributions of atg-triplets of complete RNA
sequences of Dengue virus-3 (rows 11-15 –(A) and 16-18
–(B) Table 3, Appendix 1).
Figure 11A shows the atg-distributions of
two Gabon-originated Dengue viruses that can
relate to predecessors of other Dengue viruses
of this family. Figure 11B presents the atg-
distributions of complete RNA sequences for
five Dengue virus-4 samples. They have
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individual and statistical differences with the
above-considered Dengue viruses.
Fig. 11: Distributions of atg-triplets of complete RNA
sequences of Dengue virus-3 (rows 19, 20 - (A), Table 3,
Appendix 1) and Dengue-4 (rows 21-25 - (B), Table 4,
Appendix 1).
A consolidated plot of all atg-curves of the
Dengue RNAs studied here is shown in Figure
12. There is a substantial divergence of these
trajectories in agreement with the mutation rate
of this virus being relatively strong. For
instance, this relative difference
1,25
y
estimated at
10000
i
x
is 14.1%.
Fig. 12: Consolidated picture for all 25 Dengue virus
arbitrary-chosen samples studied in this paper. The
numbers of virus atg-curves correspond to Table 3,
Appendix 1.
3.4. Analysis of atg-walks of Complete
Genome Sequences of the Ebola Virus
There are four strains of the Ebola virus known
worldwide, although many other mutations of
this virus can be found. Like the Dengue virus,
Ebola shows instability and an increased rate of
mutations. Initially, the infection was registered
in South Sudan and the Democratic Republic of
the Congo, and it spreads due to contact with
the body fluids of primates and humans. This
fever is distinguished by a high death rate (from
25% to 90% of the infected individuals). A
recent comprehensive review of the genomics
of this virus can be found in Ref. [77].
The Ebola virus RNA consists of 19 000
nucleotides and more than three hundred atg-
triplets. Figure 13A shows four sequences of
this virus belonging to the EBOV strain
registered in Zaire and Gabon. Three of them
are very close to each other, but the mutant
Zaire virus (in red) has some differences from
the three others. The samples collected in Sudan
(SUDV) are closer to each other (Figure 13B),
but they have an increased number of atg-
triplets and shorter words.
Fig. 13: Distributions of atg-triplets of complete RNA
sequences of the Ebola - EBOV) virus from Zaire and
Gabon (rows 1-4 - (A), Table 4, Appendix 1) and Ebola
virus - SUDV from Sudan (rows 5-7 - (B), Table 4,
Appendix 1).
The Bombali virus, registered in Sierra
Leone, West Africa, is considered a new strain
of Ebola. The atg-distributions of the five found
in GenBank RNA sequences are different even
visually from the two reviewed above, as seen
in Figure 14A. Another Ebola strain that can be
compared with the one studied above is the
Bundibugyo (BDBV) virus, whose three atg-
distributions are shown in Figure 14B.
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Fig. 14: Distributions of atg-triplets of five complete
RNA sequences of the Ebola (Bombali) virus (rows 8-12
– (A), Table 4, Appendix 1) and three complete RNA
sequences of the Ebola Bundibugyo (BDBV) virus. See
rows 13-15 – (B), Table 4, Appendix 1.
The calculated 15 distributions are
consolidated in Figure 15 to compare all four
strains, where, instead of points, the results are
represented by thin curves to make these
distributions more visible. Here, the tendency of
atg-curves to divergence and form clusters is
seen. For instance, the relative difference of
strains
1,15
y
is estimated at around 9%.
Fig. 15: Consolidated representation of 15 atg-
distributions of four strains of the Ebola virus. Black
color – EBOV; Green color – SUDV; Violet color –
Bombali; Red color – BDBV. The numbers of virus atg-
curves correspond to Table 4, Appendix 1.
It follows that the atg-walk is an effective
visualization tool sensitive to the viral RNA
mutations coupled to the number of codons and
introns and word size variations. It allows the
detection of viruses with essentially unstable
genomes distinguished by their increased
deviation of atg-walks and their fractal
properties.
3.5 Statistical Characterisation of atg-walks:
Calculating, Mapping, and Processing of the
Inter-atg Distance Values
In this research, after applying the tool FracLab
(See Section 2.2.3), it was discovered that all
studied genomic sequences of the SARS CoV-
2, MERS CoV, Dengue, and Ebola viruses have
fractality in their word-length distributions. The
results on this matter are placed in columns 6 of
Tables 1-4, Appendix 1.
Figure 16A shows the fractal dimension
values
F
d
of 37 genome sequences of SARS
CoV-2 viruses. Figure 16B gives this parameter
for 20 MERS coronavirus samples. Calculations
show that the maximal normalized deviation
(relative difference) of
F
d
is only 1.8% for
SARS and 2% for MERS viruses, i.e., these
results are in accordance with the conclusion on
the stability of these viruses towards severe
mutations with the codon-length and codon-
content perturbations.
Fig. 16: Fractal dimensions
F
d
of word-length
,1
atg
ii
l
distributions of 37 complete genome sequences of the
SARS CoV-2 (A) and 25 complete genome sequences of
the MERS CoV (B) viruses. The number of samples
corresponds to Tables 1 and 2 of Appendix 1.
The Dengue virus has five families and 47
strains; they have different atg-distributions and
fractal dimensions. According to the fractal
calculations, some strains are close to each
other (Figure 17).
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Fig. 17: Fractal dimensions
F
d
of word-length
,1
atg
ii
l
distributions of 25 complete genome sequences of the
Dengue 1-4 viruses and their strains (Table 3, Appendix
1).
At the same time, the maximal relative
deviation is around 22% estimated for all
studied samples, and it agrees that this virus is
unstable from strain to strain.
The same conclusion is evident in Figure 18,
where the fractal dimensions of several studied
strains of the Ebola virus are given with a
maximal relative deviation of around 18%.
Fig. 18: Fractal dimensions
F
d
of word-length
,1
atg
ii
l
distributions of 15 complete genome sequences of the
Ebola virus strains (Table 4, Appendix 1).
4. Discussion
The research on the RNAs and DNAs of viruses
and cellular organisms is a highly complex
problem because of the many nucleotides of
these organic polymers, unclear mechanisms of
their synthesis, and pathological mutation
consequences for host organisms. Although
many mathematical tools have been developed,
new studies are exciting and can be fruitful.
In this paper, the viral RNAs were studied
using a novel algorithm based on exploring the
RNA patterns of arbitrary length. One of the
operations of this algorithm is the numerical
mapping of RNA characters, which is
performed by calculating the Hamming distance
between the preliminary binary-expressed
queries and RNA symbols. This allows
fulfilling these steps approximately twice as fast
regarding the operations with real numbers [45].
The results of the application of this algorithm
are verified by comparing them with complete
RNA sequences. Considering this algorithm can
search arbitrary-length patterns, the trajectories
of separate symbols can be combined with the
atg-walks for multi-scale plotting and RNA
analysis, as shown in this contribution.
The mentioned atg-triplets compose
relatively stable 1-D distributions called the
RNA schemes in this paper. These distributions
have been studied using our algorithm applied
to arbitrary-chosen complete RNA sequences of
the SARS CoV-2, MERS CoV, Dengue, and
Ebola viruses.
The following properties of virus RNAs have
been found in our research and not seen earlier:
1. Stability of atg-schemes towards intra-
family mutations when the geometry of
atg-curves is only slightly distorted
according to the estimates of the relative
difference of curve points (Section 3)
2. The highly compact atg-curve sets of
the SARS CoV-2 and MERS CoV
viruses, despite their continuous
mutation (to the date of submission),
estimated visually and quantitively
calculating the relative difference of atg-
coordinates (δy=1-1.6%), see Sections
3.1 and 3.2
3. More substantial divergence (δy=9-
14%) of atg-curves of the Dengue and
Ebola viruses in comparison to SARS
CoV-2 and MERS CoV species
(Sections 3.3 and 3.4 and 3.1, 3.2)
4. A visually found tendency towards
clustering the atg-curves in the limits of
one virus family (Ebola case, Section
3.4)
5. Distribution of single RNA’s symbols
and atg-triplets according to the random
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fractal Cantor rule (Section 2.2.2.1)
6. Possible global correlation of the inter-
triplet distances due to this fractality
(Section 2.2.3)
7. Correlation of dispersion of fractal
dimension values of atg-distributions
with the instability of viruses (Section
3.5)
5. Conclusion
In this paper, the visual and quantitative
analyses of viral RNAs have been performed
using a novel algorithm to calculate the RNA
pattern positions in the studied sequences. A
part of this code uses binary symbols of RNA
nucleotides for accelerated search. The
algorithm allows more effective genomic
studies by building 1-D distributions of
different patterns and combining these sets on a
single multi-scale plot.
The proposed techniques were applied to
analyze the SARS CoV-2, MERS CoV,
Dengue, and Ebola viruses. The 1-D
distributions of atg-triplets (atg-schemes of
RNAs) were calculated and plotted for these
species. The levels of stability of these
distributions have been estimated visually and
numerically by calculating the divergence of
triplet curves and deviation of word fractal
dimensions of RNAs.
The main finding of this work is that the
level of clustering of atg-curves is with the
degree of stability of RNAs shown comparing
the SARS-CoV-2 and MERS species and the
Dengue and Ebola viruses. In addition, the rank
of clustering is coupled with the deviation of
fractal dimensions of codon-length
distributions, which is more significant for the
unstable Dengue and Ebola viruses mentioned.
It may be found engaging in further study of the
mutation of viruses and building their
phylogenetic trees.
The developed approach is with the study of
the RNA words, their lengths, and fractal
properties of the word-length distributions. It
can be applied in the research of mammalian
DNAs where the gene length is the evolutionary
dynamic and partly defines the gene expression
level [78].
In addition, recent studies show that with
aging, an imbalance of short and long genes in
the transcriptome occurs, and the research of
these phenomena by our algorithm may play an
important role in the development of anti-aging
treatments [79].
Thus, even quantitative and qualitative
studies and modeling of short virus genes can
be helpful in the genetics of more complicated
mammalian DNAs consisting of billions of
nucleotides.
Abbreviations
RNA: Ribonucleic acid; DNA:
Deoxyribonucleic acid; SARS CoV-2: Severe
Acute Respiratory Syndrome Coronavirus 2;
cDNA: complementary DNA; GISAID: Global
Initiative on Sharing All Influenza Data; NP:
nondeterministic polynomial; UTF-8: Unicode
Transformation Format-8 bit; US-ASCII:
American Standard Code for Information
Interchange; MERS CoV: Middle-East
Respiratory Syndrome-related Corona Virus.
Acknowledgments
The authors thank the GenBank® [39] and
GISAID [40] genetic data banks, and all
researchers placed their genomic sequences in
them. The online text processing service of
https://onlinetexttools.com/ is appreciated.
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Appendix 1. Results of statistical characterization of complete genetic sequences of the SARS CoV-2,
MERS CoV, Dengue, and Ebola viruses
Table 1. Severe acute respiratory syndrome coronavirus 2, (GenBank, GISAID), atg-walk
#
GenBank or GISAID Virus Name,
Clade, Lineage, Registration Year,
Sequencing Technology
Number of
Nucleotides in
the Sequence
Number
of atg-
Triplets
in the
sequence
Word
Median
Length
RMS
Word
Length
Fractal
Dimension of
the Word-
length
Distribution
1
2
3
4
5
6
1
GenBank: MN988668.1, Severe
acute respiratory syndrome
coronavirus 2 isolate 2019-nCoV
WHU01, Wuhan, China, 2020,
Illumina
29881
725
29
57.93
2.17
2
hCoV-19/Japan/NGY-NNH-
075/2021, GR, B.1.1.64, Illumina
MiSeq, Sanger
29848
722
29
58.03
2.17
3
hCoV-19/India/ILSGS00925/2021,
G, (Delta) B.1.617.2, Illumina
NextSeq550
29782
723
28.05
57.77
2.16
4
hCoV-19/South
Korea/KDCA3504/2021, GH,
B.1.497, Illumina Miseq
29901
722
29
57.96
2.17
5
hCoV-19/Taiwan/TSGH-34/2020, S,
A.1, Illumina NovaSeq4000
29903
724
29
57.79
2.17
6
hCoV-
19/bat/Cambodia/RShSTT182/2010,
A.1, (bat virus), 2021, Illumina
NextSeq
29787
730
29
55.81
2.17
7
hCoV-19/Austria/CeMM3224/2021,
GR, B.1.1.244, Illumina NovaSeq
29782
721
30
59.03
2.16
8
hCoV-19/England/205341113/2020,
GV, B.1.177.54, Illumina NextSeq
29862
721
29
57.97
2.17
9
hCoV-19/Ireland/D-NVRL-
e84IRL94434/2021, GV, B.1.177,
Illumina
29523
719
29
59.56
2.17
10
hCoV-19/Netherlands/UT-RIVM-
13868/2021, GH, B. 1.160,
Nanopore MinION
29782
720
28
58.17
2.16
11
hCoV-19/Norway/0179/2021, GH,
B.1.36, Nanopore GridIon
29782
723
28
57.88
2.15
12
hCoV-19/Russia/IVA-CRIE-
L188N0202/2021, GR, B.1.1.317,
Illumina
29735
720
29
57.77
2.17
13
hCoV-19/Spain/RI-IBV-
99016064/2021, GV, B.1.221,
Illumina MiSeq
29865
719
29
59.56
2.17
14
hCoV-19/Brazil/RS-
00674HM_LMM52649/2020, GR,
B.1.1.33, Illumina Miseq
29867
719
29
58.31
2.17
15
hCoV-19/Canada/ON-S2383/2021,
GH, B. 1.36.38, Illumina MiniSeq
29830
722
29
57.89
2.16
16
hCoV-19/Mexico/CMX-INER-
0222/2020, G, B.1.551, Illumina
NextSeq
29885
724
29
57.83
2.17
17
hCoV-19/USA/TX-HHD-
2102044112/2021, GR, B.1.1.244,
29819
720
29
58.10
2.17
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Illumina MiSeq
18
hCoV-19/USA/CA-LACPHL-
AF00513/2021, GH, B.1.429,
Illumina MiSeq
29844
723
29
57.86
2.17
19
hCoV-19/South Africa/KRISP-
K004540/2020, GR, B.1.1.56,
Illumina MiSeq
29851
722
29
57.90
2.17
20
hCoV-19/Canada/ON-NML-
254107/2021, GR, BA.1 (Omicron),
Oxford Nanopore GridION
29685
718
29
57.73
2.17
21
hCoV-19/England/MILK-
2D6B000/2021, GRA, BA.2
(Omicron), Illumina NovaSeq
29724
725
28.5
57.48
2.16
22
hCoV-19/USA/CA-CDC-
LC0934366/2022, GRA, BQ.1,
PacBio Sequel II
29642
718
28
57.31
2.19
23
hCoV-19/USA/CA-CDC-
LC0933659/2022, GRA, BQ.1,
PacBio Sequel II
29668
725
28
57.5
2.16
24
hCoV-19/Canada/QC-
L00549819001/2022, GRA, BQ.1,
Illumina NextSeq
29719
725
28
57.5
2.16
25
hCoV-
19/Malaysia/IMR/CV05212/2022,
GRA, BQ.1.23, Nanopore GridION
29636
725
28
57.5
2.17
26
hCoV-19/Ireland/CO-CUH-
S22C0165/2022, GRA, BQ.1,
Nanopore MinION
29646
723
28
57.4
2.18
27
hCoV-19/Canada/QC-
L00548493001/2022, GRA, BA.4,
Oxford Nanopore PromethION
29709
724
29
57.5
2.16
28
hCoV-19/Guatemala/7817-
LNS/2022 GRA, BA.4,
Illumina MiSeq
29714
724
29
57.48
2.16
29
hcov-19/USA/UT-UPHL-
221115393771/2022, GRA, BA.4.6,
Illumina NovaSeq 6000
29734
723
28.5
57.68
2.16
30
hCoV-19/Germany/RP-USAFSAM-
S20372/2022, GRA, BA.5,
Illumina_NextSeq_Mid
29693
722
29
57.66
2.15
31
hCoV-19/Russia/MOW-CRIE-
89961/2022, GRA, BA.5, Oxford
Nanopore
29646
719
29
57.65
2.17
32
hCoV-19/Norway/OUS-26253/2022,
GRA, BA.5.1, Illumina Swift
Amplicon SARS-CoV-2 protocol at
Norwegian Sequencing Centre
29605
719
29
57.61
2.17
33
hCoV-19/USA/CA-CDPH-
FS48102807/2022, GRA, BA.5.1.1,
Element Biosciences
29607
723
29
57.36
2.17
34
hCoV-19/USA/NY-
NYULH9985/2022, GRA, XBB.1.5,
Amplicon (Illumina), Illumina
NovaSeq
29773
726
29
57.44
2.17
35
hCoV-19/USA/NJ-PHEL-
V22054945/2022, GRA, XBB.1.5,
Oxford_Nanopore
29649
722
29
57.48
2.17
36
hCoV-19/Iceland/L-3254/2022,
GRA, XBB.1.5, Illumina MiSeq
29669
725
28.5
57.47
2.16
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37
hCoV-19/Ireland/CO-CUH-
S23C0065/2023.txt, GRA, CH.1.1,
Nanopore MinION
29652
722
29
57.49
2.17
Table 2. The Middle East respiratory syndrome-related coronavirus, (GenBank), atg-walk
#
GenBank Virus Name and
Accession Number, Registration
Year, Sequencing Technology
Nucleotides
Number
Number
of atg-
Triplets
Word
Median
Length
RMS
Word
Length
Fractal
Regularization
Dimension of
the Word-
length
Distribution
1
2
3
4
5
6
1
MF598617.1, Middle East
respiratory syndrome-related
coronavirus strain
camel/UAE_B25_2015, United
Arabian Emirates, AE, 2017,
Illumina; Sanger dideoxy
sequencing
30123
712
30
58.8
2.30
2
MF598595.1, Middle East
respiratory syndrome-related
coronavirus strain
camel/UAE_B2_2015, United
Arabian Emirates, 2017,
Illumina; Sanger dideoxy
30123
709
30
59.04
2.30
3
NC-019843.3, Middle East
respiratory syndrome-related
coronavirus isolate HCoV-
EMC/2012, Saudi Arabia, 2020,
Sanger dideoxy
30119
717
30
58.48
2.30
4
KY673148.1, Middle East
respiratory syndrome-related
coronavirus strain
Hu/Oman_50_2015, 2017,
Sanger dideoxy
30123
714
29
58.74
2.30
5
KT225476.2, Middle East
respiratory syndrome coronavirus
isolate MERS-
CoV/THA/CU/17_06_2015,
Oman/Thailand, 2017, Sanger
dideoxy
29809
703
30
59.03
2.25
6
MG923479.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-CoV
camel/Nigeria/NV1712/2016,
2018, Sanger dideoxy
29455
701
30
58.08
2.24
7
MK967708.1, Middle East
respiratory syndrome-related
coronavirus isolate
Merscov/Egypt/Camel/AHRI-
FAO-1/2018, 2019, CLC
genomic workbench
30106
711
30
58.05
2.30
8
MT361640.1, Mutant Middle East
respiratory syndrome-related
coronavirus clone MERS-CoV
YKC, South Korea, 2021,
sequencing technology is
described in [76]
30136
710
30
58.90
2.30
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9
KT326819.1, Middle East
respiratory syndrome coronavirus
strain MERS-
CoV/KOR/KNIH/001_05_2015,
South Korea, 2017, Illumina and
Sanger dideoxy
29995
711
30
58.86
2.30
10
MK129253.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-
CoV/KOR/KCDC/001_2018-
TSVi, South Korea, 2019, Sanger
dideoxy
30150
712
30
58.81
2.29
11
OL622035.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-
CoV_Riyadh_2016, 2021, Oxford
Nanopore
29994
709
30
59.03
2.29
12
OP712625.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-
CoV/dromedary
camel/Egypt/NC4714/2016,
2022, Illumina
30108
719
30
57.88
2.3
13
MH734114.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-CoV
camel/Kenya/C1215/2018, 2018
30033
721
29.5
58.05
2.27
14
MG923468.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-CoV
camel/Ethiopia/AAU-EPHI-
HKU4458/2017, 2018
30091
722
30
57.65
2.3
15
KJ361503.1, Middle East
respiratory syndrome coronavirus
isolate Hu-France -
FRA2_130569-
2013_Isolate_Sanger, 2014,
Sanger dideoxy sequencing
30040
710
29
59.18
2.27
16
KM210277.1, Middle East
respiratory syndrome coronavirus
isolate England/4/2013, complete
genome, 2014, Sanger dideoxy
sequencing
30031
712
30
58.81
2.27
17
KF958702.1, Middle East
respiratory syndrome coronavirus
isolate MERS-CoV-Jeddah-
human-1, 2013, Sanger dideoxy
sequencing
29851
709
29
58.67
2.25
18
OP654179.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-
CoV/dromedary
camel/Egypt/NC270-P9/2015,
2022, Illumina
30131
711
30
58.87
2.3
19
MW086535.1, Middle East
respiratory syndrome-related
coronavirus isolate MERS-
CoV/JC32/Ramtha, 2020,
Illumina
29825
707
29.5
58.72
2.25
20
MZ268405.1, Middle East
30106
718
30
58.19
2.3
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E-ISSN: 2224-266X
345
respiratory syndrome-related
coronavirus isolate MERS-
CoV/Camel/Kenya/HKU-
CAC10200/2020, 2021, Illumina
Table 3. The Dengue virus, (GenBank), atg-walk
#
GenBank Virus Name,
Registration Year, Sequencing
Technology
Nucleotides
Number
Number
of atg-
Triplets
Word
Median
Length
RMS
Word
Length
Fractal
Regularization
Dimension of
the Word-
length
Distribution
1
2
3
4
5
6
1
KY672944.1, Dengue virus 1
isolate DENV-
1/China/YN/YNH22 (2013),
2019, Sanger dideoxy
10709
299
23
47.74
2.36
2
KY672937.1, Dengue virus 1
isolate DENV-
1/China/YN/DGRL-6(2014),
2019, Sanger dideoxy
10738
294
23
50.02
2.33
3
MW386865.1, Dengue virus 1
isolate YNBN04, China, 2020,
Sanger dideoxy
10742
289
24
50.81
2.36
4
MG560269.1, Dengue virus 1
isolate
P1253/China/GD/CZ/2014,
2018, Sanger dideoxy
10583
298
23
47.55
2.35
5
MG560267.1, Dengue virus 1
isolate
P1258/China/GD/CZ/2014,
2018, Sanger dideoxy
10583
299
23
47.22
2.35
6
MN566112.1, Dengue virus 2
isolate New Caledonia-2018-
AVS127, 2020, Illumina
10722
267
32
52.24
2.4
7
KY672955.1, Dengue virus 2
isolate DENV-
2/China/YN/15DGR65(2015),
2019, Sanger dideoxy
10723
273
28
52.77
2.44
8
KY672954.1, Dengue virus 2
isolate DENV-
2/China/YN/JH1516(2015),
2019, Sanger dideoxy
10665
271
29
51.50
2.48
9
MK268692.1, Dengue virus 2
isolate DENV-2/TH/1974,
Thailand, 2019, Sanger dideoxy
10721
274
28
52.67
2.45
10
MH069499.1, Dengue virus 2
strain DENV-
2/VE/IDAMS/910105,
Venezuela, 2018, Illumina
10712
275
28
52.84
2.49
11
MN018389.1, Dengue virus 3
isolate D17011, China, 2020,
Sanger dideoxy sequencing
10708
272
28
55.46
2.57
12
NC_001475.3, Dengue virus 3,
Sri Lanka, 2019, Illumina
10707
273
27
55.05
2.58
13
KY863456 .1, Dengue virus 3
isolate 201610225, Indonesia,
2017, IonTorrent, Sanger
10707
278
28
52.84
2.5
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.35
Alexandra Belinsky, Guennadi A. Kouzaev
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346
dideoxy sequencing
14
MH544649.1, Dengue virus 3
isolate
449686_Antioquia_CO_2015,
Colombia, 2018, Illumina;
Sanger dideoxy sequencing
10707
273
28
52.84
2.49
15
MH823209.1, Dengue virus 3
isolate SMD-031, Indonesia,
2019, Illumina
10707
272
28
52.84
2.46
16
LC379197.1, Dengue virus 3
strain SYMAV-17/Gabon/2017
genomic RNA, 2019, Illumina
10641
271
29
52.85
2.06
17
KY921907.1, Dengue virus 3
isolate SG(EHI)D3/15095Y15,
2017, Singapore, Sanger dideoxy
sequencing
10667
266
29
53.58
2.09
18
KF041255.1, Dengue virus 3
isolate D3/Pakistan/55505/2007,
2013, Sanger dideoxy
sequencing
10675
268
29
53.55
2.07
19
LC379196.1, Dengue virus 3
strain SYMAV-09/Gabon/2016
genomic RNA, 2019, Illumina
10663
273
29
53.64
2.53
20
LC379195.1, Dengue virus 3
strain SYMAV-07/Gabon/2016
genomic RNA, 2019, Illumina
10663
273
29
53.64
2.53
21
KJ579245.1, Dengue virus 4
strain DENV-
4/MT/BR23_TVP17909/2012,
Brazil, 2020, Illumina
10649
273
26
53.12
2.09
22
MG272274.1, Dengue virus 4
isolate D4/IND/PUNE/IRSHA-
FG-03 (S-49), complete genome,
India, 2018, Ion Proton System
10652
270
27
53.07
2.07
23
KY672960.1, Dengue virus 4
isolate DENV-
4/China/YN/15DGR394 (2015),
2019, Sanger dideoxy
10661
276
26
52.63
2.09
24
KX224312.2, Dengue virus 4
isolate SG(EHI)D4/02990Y14,
Singapore, 2017, Sanger dideoxy
sequencing
10652
275
27
52.21
2.09
25
MG272272.1, Dengue virus 4
isolate D4/IND/PUNE/IRSHA-
FG-01 (1028), India, 2018, Ion
Proton System
10652
272
27
54.57
2.09
Table 4. The Ebola Virus, (GenBank), atg-walk
#
Genbank Virus Name, Registration Year,
Sequencing Technology
Nucleotide
Number
Number
of atg-
Triplets
Word
Median
Length
RMS
Word
Length
Fractal
Dimension
of the Word-
length
Distribution
1
2
3
4
5
6
1
MG572235.1, Zaire ebolavirus isolate
Ebola virus/H.sapiens-
tc/COD/1995/Kikwit-9510621, Zaire,
2019, PacBio; Illumina
18957
329
40.5
85.29
2.03
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.35
Alexandra Belinsky, Guennadi A. Kouzaev
E-ISSN: 2224-266X
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2
KU174137.1, Mutant Zaire ebolavirus
isolate Ebola virus/H.sapiens-
rec/COD/1976/Yambuku-Mayinga-
eGFP-BDBV_GP, Zaire, 2019, Illumina
19774
339
41.5
84.02
2.06
3
KY786025.1, Ebola virus strain Ebola
virus/M.fascicularis-
wt/GAB/2001/untreated-CCL053D5,
Gabon, 2018, IonTorrent
18871
327
40.5
85.06
2.03
4
KY785936.1, Ebola virus strain Ebola
virus/M.fascicularis-
wt/GAB/2001/100mg-CA470D5, Gabon,
2018, IonTorrent
18871
327
40.5
85.06
2.03
5
MH121162.1, Sudan ebolavirus isolate
Ebola virus/H.sapiens-
tc/Sudan/1976/Boniface-R4142L, 2019,
Illumina
18831
345
37.5
77.79
2.2
6
MK952150.1, Sudan ebolavirus isolate
Ebola virus/H.sapiens-
wt/SSD/1976/Maridi-BNI/DT, South
Sudan, 2020, Illumina
18847
345
37
77.84
2.2
7
MH121169.1, Sudan ebolavirus isolate
Ebola virus/H.sapiens-
tc/Sudan/2004/Yambio-HCM/SAV/017,
2019, PacBio; Illumina
18849
345
37
77.84
2.2
8
NC_039345.1, Bombali ebolavirus
isolate Bombali ebolavirus/Mops
condylurus/SLE/2016/PREDICT_SLAB
000156, Sierra Leone, 2018, Sanger
dideoxy, Illumina
19043
325
36
84.24
2.38
9
MW056492.1, Bombali ebolavirus
isolate X030, Kenya, 2020, Illumina
19025
326
36
84.11
2.37
10
MF319186.1, Bombali ebolavirus isolate
Bombali virus/C.pumilus-
wt/SLE/2016/Northern Province-
PREDICT_SLAB000047, Sierra Leone,
2019, Sanger dideoxy
19043
324
36
85.27
2.36
11
MK340750.1, Bombali ebolavirus isolate
B241, Kenya, 2019, Illumina
19025
328
36
83.59
2.36
12
MW056493.1, Bombali ebolavirus
isolate Z153, Kenya, 2020, Illumina
19025
332
36
81.46
2.41
13
MK028856.1, Bundibugyo ebolavirus
isolate Ebola virus/H.sapiens-
tc/Uganda/2007/Bundibugyo-
200706291, 2019, PacBio; Illumina
18940
325
39
84.62
2.01
14
MK028834.1, Bundibugyo ebolavirus
isolate Ebola virus/H.sapiens-
tc/Uganda/2007/Bundibugyo-R4386L,
Uganda, 2019, PacBio; Illumina
18917
325
39
84.62
2.01
15
MK028835.1, Bundibugyo ebolavirus
isolate Ebola virus/H.sapiens-
tc/Uganda/2007/Bundibugyo-
200706291, 2019, PacBio; Illumina
18936
325
39
84.63
2.01
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.35
Alexandra Belinsky, Guennadi A. Kouzaev
E-ISSN: 2224-266X
348