Momentum Distribution of SARS-CoV-2 Sequences on Variant Maps

Applying momentum measures and probability distributions, a visual scheme is proposed in this paper to transfer SARS-CoV-2 genomes to two distinct maps to be the A6 module of the MAS. Based on collected datasets of SARS-CoV-2 on different countries as examples, relevant characteristics and correlation of probability distributions are described. In visualization of two patterns under momentum measurement, there are two visual forms of Pattern 1 and Pattern 2. Two forms are generated from the same sequence on different focuses. Pattern 1 describes the momentum distribution of a segment accounting for whole genomes, and Pattern 2 visualizes the differences in momentum distributions on various segments. This is a new type of analysis measurement to explore complex connections and differences among sets of SARS-CoV-2 genomes for their availability.


Momentum
In classical mechanics, momentum is quantified as the product of the mass and velocity of an object [1].Globally, the momentum of an object refers to the tendency of the object to keep moving in its moving direction [2].Momentum is actually a corollary of Newton's first law.Momentum is a conserved quantity, which means that the sum of momentum in a closed system cannot be changed.Conservation of momentum is implicit in Newton's law, but it still holds in special relativity [3], and (generalized) momentum holds in electrodynamics [4], quantum mechanics [5,6], quantum field theory [7], and general relativity.

Variant Construction and SARS-CoV-2
The SARS-CoV-2 sequences are composed of four symbols on {A, T, G,C}.The classic DNA double helix [8] structure makes the primary matching structure of the DNA sequence and the complementary structure of A & T and G & C the first level of the symmetric relationship [9].
The variant construction includes logic, measurement, and visualization models [10][11][12], which aim to analyze 0-1 sequences as vectors.Four meta symbols [13,14] were used for the variant construction to describe a natural mapping between DNA and variant data sequences [14].
In variant systems of DNA analysis, it is significant for frontier researchers to focus on higher level interpretation.Based on the variant method, their complex interactive characteristics could be organized as different visual maps systematically.

Variant Mapping
Variant mapping is proposed to address spatial changes in signal phase to visualize different DNA sequence properties and has many applications [15][16][17][18].Variant maps are a perfect approach to handle four symbols as meta-structures to process DNA sequences [19,20], ECG signals [21],and encrypted sequences [22].In this paper, the connection of momentum and variant mapping is proposed to use a fixed segment structure to calculate the momentum of SARS-CoV-2 sequences and represent various distribution maps.

Aim of The Study
The aim of this paper is to propose the variant map under momentum measurement as a calculation tool to analyze SARS-CoV-2 sequences from various countries.The momentum measurement is designed to use four meta symbols as input, select a fixed number of segments m, calculate their first-order origin momentum and second-order central momentum, and output variant maps of two patterns.The larger the value of m is, the greater the number of graph blocks.It is not convenient to view the relationship between segment and four meta symbols, and poor visualization results are not conducted to analyze the interactions.In the following, m = 10 is selected as the variable value mapping of the first-order origin momentum and the second-order central momentum of the samples.
Pattern 1 is composed of five rings and 10 blocks, which are connected to each other.The outermost ring is a standard reference ring mapped by momentum measure and probability distribution, and the remaining four rings are mapped SARS-CoV-2 sequences.From the outer to the inner are momentum of {A, T, G,C}.Different rings illustrate each distribution of momentum (A), momentum (T), momentum (G), and momentum (C) and are compared with the standard reference ring.Different blocks reversely correspond to the segments of 0 to 9, and the baseline is at the beginning of segment 0, showing the whole distribution of momentum {A, T, G,C}.A block can be compared with the distributions of momentum (A), momentum (T), momentum (G), and momentum (C) in the same segment.
Pattern 2 contains four closed curves with different colors, and each curve represents the probability distribution of momentum (A), momentum (T), momentum (G), and momentum (C).It shows the difference distribution of momentum on the same segment.The coincident curves indicate that they have the same value of momentum in this segment, but the curves are sparse, indicating that the values of momentum vary greatly.

Relevant Parameters
First order origin momentum of A momentum T 1 ( j) num(T) ( j) /M ( j) First order origin momentum of T momentum G 1 ( j) num(G) ( j) /M ( j) First order origin momentum of G momentum C 1 ( j) num(C) ( j) /M ( j) First order origin momentum of C Table 1: First origin momentum measurement Second order central momentum of C

P:
The proportion of a base P1: P(momentum 1)

Datasets
All datasets used in this paper were from GISAID [23], one of the largest influenza and new coronavirus data platforms in the world.A total of 1934 SARS-CoV-2 samples are randomly selected from 10 countries, including the USA, France, Australia, Japan, Italy, China, Chile, Germany, Singapore, and Canada, as shown in Figure 1.

SARS-CoV-2 sequences from various countries
From Fig. 2 and Fig. 3 visual observation, when m = 10, the map has shown the better effects.

More details
From Fig. 4 and Fig. 5, we can visualize the differences and similarities among the various countries on variant maps.Fig. 6 and Fig. 7 show their similarities under Pattern 1 and Pattern 2.

Conclusion
This paper presents a new scheme to use momentum measurement and probability distribution to describe SARS-CoV-2 sequence variant maps under two patterns.Pattern 1 and Pattern 2 can visualize four meta symbols as momentum distributions with visual characteristics.It is advantageous to perform effective analysis and visualization of sequences from different countries.Further explorations are required on higher momentum and method of calculation.

Fig. 1 :
Fig. 1: Experimental data from 10 countries Fig. 4a-j are shown for SARS-CoV-2 sequences form the USA, France, Australia, Japan, Italy, China, Chile, Germany, Singapore, and Canada under Pattern 1, and Fig. 5a-j are shown based on Pattern 2.

Figure 6 Similarities of Pattern 1
Figure 6

Figure 7 Similarities of Pattern 2
Figure 7

Table 2 :
Second order central momentum measurement