Parametric Multispectral Mappings and Comparative Genomics

Abstract

This article describes new algorithms that allow for viewing genetic sequences in the form of their multispectral images. We presented examples of the construction of such mappings with a demonstration of the practical problems of comparative genomics. New DNA visualization tools seem promising, thanks to their informativeness and representativeness. The research illustrates how a novel sort of multispectral mapping, based on decomposition in several parametric spaces, can be created for comparative genetics. This appears to be a crucial step in the investigation of the genetic coding phenomenon and in practical activities, such as forensics, genetic testing, genealogical analysis, etc. The article gives examples of multispectral parametric sets for various types of coordinate systems. We build mappings using binary sub-alphabets of purine/pyrimidine and keto/amino. We presented 2D and 3D renderings in different characteristic spaces: structural, integral, cyclic, spherical, and third-order spherical. This research is based on the method previously developed by the author for visualizing genetic information based on new molecular genetic algorithms. One of the types of mappings, namely two-dimensional, is an object of discrete geometry, a symmetrical square matrix of high dimension. The fundamental properties of symmetry, which are traced on these mappings, allow us to speak about the close connection between the phenomenon of genetic coding and symmetry when using the developed mathematical apparatus for representing large volumes of complexly organized molecular genetic information.

1. Introduction

Algorithms for processing big data have been recently intensively developed. This equally applies to the analysis of big genetic data. RNA and DNA molecules can contain several thousand to 10⁹ nucleotides, which requires in-depth computer analysis. For this, machine learning algorithms are used, and new specialized genetic algorithms and methods for visualizing the nucleotide composition are being developed. These studies are at the crossroads of diverse areas and are interdisciplinary, since they combine both the methods of molecular genetics, the systems of artificial intelligence, and data mining together with technical means for analyzing biological data. Simultaneously, there is currently no unified standard algorithm for processing and visualizing big genetic data. Thus, studies aimed at finding such unified algorithms of visualization of genetic data are relevant. This work is a development of the earlier stated concept of genometric analysis and visualization described in publications [1,2,3].

2. Literature Review

Recently, a number of publications have appeared on the technical instruments of visualization and processing algorithms of genetic information for bioengineering and computer science tasks. There is a trend towards the development of new algorithms for the visualization of genetic data. When using vitrified samples and cryo-electron microscopy, it is possible to observe the shape of supercoiled DNA molecules in solutions [4]. For a comprehensive range of applications in materials science, optics, plasmonics, molecular patterning, and nanomedicine, researchers can now choose the best production techniques and design paradigms for creating unique DNA nano-objects and software from multiple options [5]. Three groups of markers—short tandem repeats, single nucleotide polymorphisms and entire mitochondrial analyses—will be crucial in the development of forensic DNA typing in the future [6].

At the same time, genetic algorithms are being developed, which are a field of applied mathematics based on biological principles. In the research [7], the authors present a hybridized model for image encryption employing a DNA sequence and a genetic algorithm. The results of the experiment confirm that the algorithm is straightforward, workable, and quick enough. The performance analysis claims that the algorithm is robust against all types of attacks, preserving improved security as a result.

The work’s [8] objective is to illustrate the value of graphical bioinformatics techniques for describing viral genomic sequences. A novel method for identifying unidentified viral strains is suggested. Through theoretical approaches for the graphical depiction of the sequences, 2D- and 3D-dynamic representations of DNA/RNA sequences, biological sequences have been represented visually. In 2D or 3D spaces, sets of material points are used to depict the sequences. A few of the theoretical methods’ applications have been briefly explored. The SARS-CoV-2 full genome sequences are shown by 2D-dynamic graphs.

Genetic algorithms are being developed, in relation to the topical problems of molecular genetics, in particular, for the diagnosis of oncological diseases [9].

According to the article [10], the DNA Features Viewer is a Python genetic sequence annotation package that enhances map readability by enabling users to customize various visual elements to address their needs.

In article [11], the authors present the design and experimental validation of a DNA tile set that can be reprogrammed to execute a wide range of 6-bit algorithms, which includes copying, sorting, palindrome recognition, multiples of three, random walking, appointing a leader, simulating cellular automata, and generating deterministic and randomized patterns. When using vitrified samples and cryo-electron microscopy, it is possible to directly see the shape of supercoiled DNA molecules in the solution [4]. For a wide range of applications in materials science, optics, plasmonics, molecular patterning, and nanomedicine, researchers can now choose the best production techniques and design paradigms for creating unique DNA nano-objects and software from a number of choices [5]. Three groups of markers—short tandem repeats, single nucleotide polymorphisms, and entire mitochondrial analyses—will be crucial in the development of forensic DNA typing in the future. Manufacturers of sequencers have included pipelines into sequencer software to make studies more practical as forensic laboratories look for the ideal pipeline of the instruments described in the article [6].

3. Materials and Methods

The method was based on the previously described algorithm [2], in which the spectral decomposition was implemented. Previously, in [3], we visualized the structural features of the RNA nucleotide composition of various coronaviruses. We also demonstrated the possibilities of visualizing the structure of the RNA nucleotide composition of different algorithms when visualizing in different metrics [1]. In all these cases, a single basic algorithm for multiscale visualization worked in parametric spaces, based on a system of orthogonal Walsh functions that reflected the features of the physicochemical structure of the four nucleotides [12].

Let us describe a clearer difference between the proposed algorithm and the algorithms described by us in a series of previous articles. Earlier, we noted that there are many options for visualization in different spaces. In this article, we will also consider various options for visualization in different spaces. However, the key difference is that we will show the spectral decomposition in these spaces. Previously, we showed the spectral decomposition in only one-dimensional space [1]. In this study, we implemented various discrete geometric algorithms to implement visualization by decomposition into multistructures (multispectral mapping). This is the subsequent step in the development of molecular genetic imaging algorithms, performed through a combination of the previously described methods.

Let us recall the main stages of the original algorithm and describe the methodology we followed. The first stage represents the choice of the scaling parameter N, which specifies the number of nucleotides in each minimal structural element (visualization point or N-plet). The subsequent step is the division of the genetic sequence into three sequences under the binary sub-alphabets system described in [1,2,3]. These sub-alphabets are a trinity of traits: purine/pyrimidine, keto/amino, and two/three hydrogen bonds. Each sub-alphabet is binary, allowing the genetic sequence to be represented as a binary string. Finally, the third stage is visualization, in which each minimal DNA fragment is displayed in a certain coordinate system, in accordance with the selected scaling parameter N. It is possible to set various options for the parametric space by choosing one or another coordinate system and principles for displaying genetic information.

As we noted above, at the first step, we divided the DNA into fragments of equal length. The length of such fragments sets the clarity of the final image and is selected experimentally. Each such fragment defines a point in some coordinate space. The coordinates of each point are given, respectively, by three sub-alphabets. Thus, in the general case, we have a three-dimensional space and its various projections.

The method of spectral decomposition boils down to the fact that, at each point, there is a certain number of nucleotides with certain properties coordinated by a system of binary sub-alphabets. For example, this may be the number of nucleoids in the N-plet with two hydrogen bonds. These numerical characteristics make it possible to decompose the visualizations into so-called spectra, which gives a more clear and detailed final mapping.

Previously, we described “structural, integral and frequency representations, in which the genetic sequence was displayed for each of the three sub-alphabets either by decimal values of the minimum fragments, or by the number of certain characters in the minimum sequence, or by frequencies, respectively.” Additionally, in [1], it was demonstrated how one can implement spectral decomposition for one-dimensional mappings. In this article, we will take a look at this step with examples in comparing two different aligned human DNAs. An example of integrated comparative visualizations of two different humans (chromosome 1) at 1D and 2D parametric spaces, shown at Figure 1.

4. Results and Discussion

For the problem of comparative genetics, the differences in the DNA of the reference genome and a sample of a real person were identified by visualization of genetic representations. We have built various visualization options. On all displays, the brightness intensity of the points on each visualization corresponds to the number of N-pleats that fall into the corresponding coordinate: the brighter the point, the higher the concentration of N-pleats.

The reference human genome is a digital sequence of nucleic acids made publicly available as a representative example of a set of genes in some “idealized” organism [13]. We sourced the initial data for mapping from the open source of the IGSR: The International Genome Sample Resource (1000 Genomes Project). The comparison was performed with the reference human genome version GRCh38.p13 and the HG00096 sample assembled on its basis (https://www.internationalgenome.org/data-portal/sample/HG00096 (accessed on 1 October 2022)) [14]. Using the bcftools tool, a consensus sequence was created for the sample.

Below are examples of multispectral parametric sets for various types of coordinate systems. To build two-dimensional mappings, we will utilize the binary features of nucleotides, such as purine/pyrimidine and keto/amino. Figure 2 shows six different renderings, three of which are 2D and three are their respective 3D generalizations. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 illustrate visualizations in various characteristic spaces: structural, integral, cyclic, spherical, and third-order spherical.

On the comparative visualizations, the green color marks the coincidence of values for the reference genome and the sample. The red color is used to mark cases when the number of hits at a given point in the reference genome is greater than in the sample genome. Ultimately, the blue color denotes cases when the number of hits at a given point in the reference genome is less than in the sample genome. As shown, this approach makes it possible to visualize the predominance of individual fragments, relative to the compared genomes.

Let us recall the spectral decomposition algorithm for the one-dimensional case (by one sub-alphabet):

• Along the ordinate axis, the space is divided into regions, the number of which is equal to the parameter N.
• Structural rendering is performed for each area.
• In each area, only those points are marked, the integral value of which corresponds to the parameter N (as noted earlier, the integral value represents the sum of units in the binary representation, according to the corresponding sub-alphabet).

For the multidimensional case (for a visualization by two or more sub-alphabets), we allow for the following generalization of this algorithm:

• For each of the coordinates, cells are laid out, the number of which is equal to the number of elements of a certain type in the minimum unique element corresponding to a certain sub-alphabet (N).
• Structural rendering is performed for each selected area.
• In each cell, a standard visualization is built, considering only those elements that satisfy their integral values to the corresponding coordinate.

Figure 4 illustrates spectral decomposition of the visualization shown at Figure 3. This visualization differs from the original one and contains additional details that characterize the spectral decomposition of the physicochemical features in the nucleotide composition of the analyzed human chromosome.

Figure 5 shows parametric multispectral mapping in two-dimensional space. The transformation is based on the mapping displayed in Figure 1 and processed according to frequencies at each N-plet. Because of the extended capabilities of multispectral decomposition, this method appears to be more powerful than those proposed earlier. This method can be considered as a step forward in the algorithms of the visualization of molecular-genetic information.

As can be seen in Figure 3, Figure 4 and Figure 5, the multispectral decomposition of genetic information is an independent implementation of the integration of structural and integral properties.

It should be noted that, for a rectangular projection in Figure 2, a three-dimensional generalization is the Sierpinski quasi-pyramid, which we demonstrated earlier in [2]. For cyclic mappings, three-dimensional projections represent objects of Riemannian geometry and are approximating spherical forms. A more comprehensive description of algorithms for constructing visualization data was presented in [1,2]. The spherical coordinate system represents a polar system, as understood in the classical sense. This system has firmly established itself as a convenient tool for studying phenomena in biology [15] and physics [16].

Next, we present comparative mappings in different cyclic spaces (Figure 6 and Figure 7).

In this research, the emphasis is on the comparative visualization of different people, namely a person with a reference genome. Each of Figure 1, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 shows visualizations of red and blue colors superimposed on each other (from two different people). Common fragments and dots are highlighted in green, and as you can see, a significant part consists of green dots, that is, they are the same for different people. All other dots that are either red or blue are indicators of difference.

It can be seen that not all visualizations are suitable for the purposes of comparative genetics. For practical purposes, the most convenient visualization is shown in Figure 3 and Figure 4. Note that, in the case of computer processing, it is possible to scale the approximation and remove individual fragments of the visualization with a detailed analysis of any fragments. Additionally, this makes the tool convenient for solving specific applied problems of visualization and comparison of the genomes of various organisms with each other.

It should be noted that, besides the DNA visualization algorithms, there are methods of DNA-sonification to facilitate the perception of big genetic data. There are scientific works on this topic. According to [17], each display point is associated with a particular note in the musical pentagram scale (or the Fibonacci-step scale [18]). This feature allows for displaying the structural features of the nucleotide composition not only graphically but also in acoustic.

Predominantly, the presented visualization and sonification methods comprise a mathematical complex of algorithms that allow for the efficient and scientific justification of the obtained visualizations, as well as to build them in various topological spaces, considering certain characteristics of nucleotides and their sets.

Simultaneously, both sonification and visualization are biologically justified, based on an objective system of genetic binary sub-alphabets [12], and they are displayed in various parametric spaces of orthogonal Walsh functions [19].

Discussing the results obtained, it should be noted that the indicated methodology of spectral DNA analysis is a fairly new scientific direction in the visualization of molecular genetic information, ergonomics of perception, and analysis of big data. The method and approaches to spectral decomposition described here are designed to improve visualization and make it clearer and more detailed, in order to facilitate the perception of large genetic data. Some visualization spaces can be described as “exotic”, and perhaps they will be useful for the purposes of scientific art by generating new discrete objects of Riemannian geometry. From the point of view of ergonomics of perception, one-dimensional and two-dimensional parametric spaces with a Cartesian coordinate system seem to be the most optimal.

In addition to the task of optimizing mental work, this direction can be considered a branch of biomathematics—a science that is devoted to the mathematical description of biological phenomena. In this case, we are talking about discrete geometry, which is a logical continuation and disclosure of the genetic algebras described in the works of S.V. Petukhov. Thus, this work is interdisciplinary in nature, which, in principle, is typical for biomathematics, in particular, for “genetic algorithms”.

5. Conclusions

The new class of spectral multistructures generated by the molecular-genetic algorithms makes it possible to display, in a more visual form, the structural features of the DNA nucleotide composition of various organisms for their visualization and comparative analysis. This lets us state that the described method of parametric multispectral mappings is the next natural step in genetic information visualization and analysis algorithms.

The presented results play a significant role in the development of methods for the ergonomics of the perception and analysis of vast amounts of genetic data in the problems of comparative and evolutionary genomics. It is expedient to introduce the described class of methods into bioinformatic software interfaces, together with machine learning systems, as well as to facilitate the perception of complex genetic information and enhancing quality of natural intelligence.

The provided examples of visualizations in different coordinate systems make it possible to visualize the fractal properties of DNA. In connection with the property of noise immunity of fractal structures, one could wonder: how can point mutations lead to significant changes in the phenotype? Let us try to explain. If the structure of the nucleotide is disrupted, then such a mutation disrupts the Walsh functions. Here, the corresponding point may fall into the “forbidden” areas of the visualizations, which correspond to empty components in 3D views. It is possible that the “butterfly effect” is involved in the mutation mechanisms, which can be studied by employing the above methods alongside the theory of the dynamic chaos [20].

The described studies demonstrate the possibilities of constructing a novel type of multispectral mapping based on decomposition in various parametric spaces for comparative genetics. This appears to mark a critical step for further research into the phenomenon of genetic coding, as well as into applied tasks such as forensics, genetic certification, genealogical analysis, etc.