Visualizations of SARS-CoV-2 Genomes on Genomic Index Maps

In this paper, comprehensive cases are visualized by { C1, C2, C3, C4 } four modules of the MAS. Four sets of SARS-CoV-2 genomes with 20, 128, 381 and 1337 cases were processed to be represented as a series of genomic index maps in four entropies: combinatorial, integrated, mean and topological entropies, respec-tively. Typical samples and explanations are described. Multiple levels of hierarchical representations are illustrated.


Genomic Index for SARS-CoV-2 Genomes
The genomic index provides unique identification for each genome to be invariant under given conditions. Based on these types of global quantitative characteristics, it is convenient for huge numbers of genomes to be located in a certain geometric region to be collected as clusters. Four papers in this special issue discuss this type of entropy quantity in separate papers: "Visualizations of Topological Entropy on SARS-CoV-2 Genomes in Multiple Regions", "2D Graphical Analysis and Visualization of SARS-CoV-2", "Cluster Analysis of Visual Differences on Pairs of SARS-CoV-2 Genomes", "Visualizations of Combinatorial Entropy Index on Whole SARS-CoV-2 Genomes".
Considering this is an extremely important research direction, it is necessary to handle this topic from a foundation level to provide additional information to explore hidden structures among this type of multiple levels of hierarchical constructions from an integrated viewpoint [3]- [9].

Useful Visualizations
In the advanced fighting fields of COVID-19, various visualization schemes and interesting structures are widely illustrated in research papers and public websites. i.e. The well-known John Hopkins website for COVID-19, the newest genomes from GISAID viral databases and dynamic Phylogeny tree on Nextstrain attracted attention from people worldwide for the newest progress and development on COVID-19 spreading situations. In relation to medical practices, intrinsic information of functional genomes provides refined differences for medical doctors and general practicers to handle complicated viral toxicities, toxicology, pharmacology and so on. A research routine on viral toxicology is shown in Fig. 1. More than 270-fold toxicities were identified among different SARS-CoV-2 genomes collected in various regions.

Brief Relationship between Phylogeny Trees and Genomic Index Maps
From a quantitative similarity viewpoint, the relationship is contained in phylogenetic structures corresponding to density distributions represented in genomic index maps shown in Fig. 2. Based on this type of correspondence, a list of transformations will be performed to show more complicated similarity information among various genomic index maps in the following parts.

Main Schemes in Processes
Following the architecture of the MAS, fifteen modules, input-output types and workflows are discussed in both papers: "A Visual Framework of Meta Genomic  Three special datasets are collected to be contained in 128, 381 and 1337 samples of SARS-CoV-2 from multiple regions worldwide to illustrate specific effects under different combinatorial and other complicated conditions. It is convenient to check each set of genomic index maps to observe specific distributions and typical structures for COVID-19 patients worldwide.

Discussion
Three sets of genomic index maps consistently provide huge invaluable information to be extracted from different sizes of SARS-CoV-2 genomes worldwide under unique quantitative invariants.

G20 and 128 Genomes on C2 and C3 Genomic Index
In Fig. 4, the first set of 20 samples is selected from the G20 regions to be represented in the (C3,C2) genomic index maps. The leftside map is shown in the G20 samples under k-mer conditions with multiple uncertainties for each sample, and the leftside map is shown in a much clearer map with distinct positions for each region by integrated mean operations. Gradually adding more genomes increased to 62 and 128, leftside maps were shown in similar clustering patterns, and 128 maps showed stronger clusters there. The integrated map is shown in more sample points, especially clustered around left-bottom parts in the map. Basic distributions between 62 and 128 were similar in shape. Two integrated maps are shown in Fig. 5, and there are significant differences between G20 and 128.
381 Genomes on C1 Genomic Index Fig. 6 to Fig. 16 provide a large number of different combinations and projections for the C1 genomic index maps. Under 2D-(A,G) projection, six 2D maps are shown in Fig. 6 to represent two integrated maps for four countries and refined regions.
In Fig. 7, both 1D-A and 1D-P projections are illustrated, leftsides on 1D-A and rightsides on 1D-G. Significant differences can be identified.
In Fig. 8, one 2D-(A,G) map and two 1D-A/1D-G projections are illustrated. There are clustering numbers associated with the relevant entropy value corresponding to whole distributions under conditions. In Fig. 9, the same maps as in Fig. 8 are arranged by their X and Y projections for convenient visual comparison, with the vertical distribution Y organized in horizontal directions as the same as the distributions in the X directions.
In Fig. 10, both 2D-(A,G) and 1D-A maps provided for four counties, and each country was associated with a 1D-A map. In Fig. 11, both 2D-(A,G) and 1D-G maps provided for four counties, and each country was associated with a 1D-G map.
In In Fig. 16, four countries and their regions were put together in one page for convenience in direct visual comparisons.

Genomes on C1 and C4 Genomic Index
Both C1 and C4 were used to process 1337 from Fig. 17 to Fig. 20 for four and eight countries, respectively.
In Fig. 17 on C1, one 2D map of all countries and five countries, the USA, China, Australia, Italy and France, were selected to show each 1D map. In Fig. 18 on C1, 2D and 1D maps of all countries and eight countries, the USA, China, Australia, Italy, Belgium, Brazil, Canada and Japan, were selected to show each 1D map.
In Fig. 19 on C4, eight countries and two countries, the USA and China were selected to show each 1D projection of various countries and regions for topological entropies, respectively. In Fig. 20 on C4, three countries, Australia, Italy and France, were selected to show each 1D projection of various regions for topological entropies.
It is interesting to see various clustering properties visualized on various genomic index maps to support further explorations for COVID-19 patients.
Genomic index schemes provide unique global invariants for genomes in general. Future activities will help people understand deeper mysteries in RNA viruses under complicated real-world environments.