# Chromatic Differentiation of Functional Mappings of the Composition of Nucleic Acids

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Relation with Chaos Game Representation (CGR)

- Set the starting point coordinate—the center of the plane (X = width/2, Y = height/2);
- Divide the X and Y values in two (see Figure 3);
- Depending on the current nucleotide, perform the following operations over the coordinates:
- A: X = X + width/2; Y = Y + height/2;
- C: X remains unaltered; Y = Y + height/2;
- G: X = X + width/2; Y remains unaltered;
- T/U: X remains unaltered; Y remains unaltered;

- Mark a point in the plane.
- Perform steps 2 to 5 over the next nucleotide; concurrently, the starting point will be the current one, i.e., the one marked in step 4.

- 1: X = X + width/2;
- 1: Y = Y + height/2;
- 1: Z = Z + depth/2;

- 0: X remains unaltered;
- 1: Y = Y + height/2;
- 0: Z = remains unaltered;

- 1: X = X + width/2;
- 0: Y remains unaltered;
- 0: Z remains unaltered;

- 0: X remains unaltered;
- 0: Y remains unaltered;
- 1: Z = Z + depth/2.

- Take the starting point coordinate–the center of the plane
- X = width/2, Y = height/2;

- Divide the X and Y values in two (see Figure 6);
- Depending on the current nucleotide, perform the following operations over the coordinates:
- A: X = X + width/2; Y = Y + height/2;
- C: X remains unaltered; Y = Y + height/2;
- G: X = X + width/2; Y remains unaltered;
- T/U: X remains unaltered; Y remains unaltered.

- Shift the center of coordinates to the obtained point. Wherein:
- width = width/2,
- height = height/2.

- Employ the next nucleotide from the N-plet and follow steps 2 through 5 until the entire nucleotide is processed. In this case, the starting point will be the current one-the new center of coordinates obtained in step 4.
- Mark a point in the plane when all nucleotides of the N-plet have been processed.
- Return the starting width and height values, and follow steps 2 to 7 for the next N-plet. In this case, the starting point will be the current one, i.e., obtained in step 6.

## 4. Results and Discussion

## 5. A Method of Chromatic Differentiation That Binds the Color to the Frequency of N-Plets

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A structural analogy between the system of binary sub-alphabets and the circle of colors (an image from [25]). In the center, we see a schematic illustration of the two cyclic sequences of the mosaic matrices, which arise because of the positional permutations in the triplets: YY

_{+,123}[CAUG] → YY

_{+,231}[CAUG] →YY

_{+,312}[CAUG] → YY

_{+,123}[CAUG] and YY

_{+,321}[CAUG] → YY

_{+,213}[CAUG] → YY

_{+,132}[CAUG] → YY

_{+,321}[CAUG]. The numbers above each matrix show a relevant type of permutation of positions in all the triplets.

**Figure 2.**The RGB color triangle and, “complementary” to it, the CMY triangle, as ultrametric subsets of complementary in terms of physicochemical characteristics of DNA of Sierpinski tetrahedra.

**Figure 3.**A schematic representation of the coordinate definition for each type of nucleotide. The dotted line shows the transformation mentioned above in step 2, and the solid line depicts the alteration outlined earlier in step 3.

**Figure 4.**Coloring each nucleotide in its color: A (red), C (blue), G (green), T/U (black). A chaos game method application is shown on the left, and on the right is an application of the Walsh method at n = 1.

**Figure 6.**A schematic representation of the calculation of the N-plet coordinate (CCG) for the chaos game method. Red dotted line—dividing the coordinate in half (step 2 of the algorithm), red solid line—modification of coordinates depending on the current nucleotide (step 3 of the algorithm).

**Figure 7.**A Comparison of visualizations obtained employing the Walsh method and that of the chaos game. For chromatic differentiation, the dot color originates from the last nucleotide in the N-plet. With relatively large N-plets, when running the chaos game, the sub-quadrants merge into a point because of the modest size of the image. Consequently, the visualizations become identical to those of Walsh. Brightness denotes the frequency of a dot: higher frequencies produce much richer colors.

**Figure 9.**Coloring at various scale parameters in the rectangular coordinate system in two and three dimensions. Two-dimensional visualizations are presented in the XY projection.

**Figure 10.**A chromatic differentiation at various scale parameters in the cyclic coordinate system in two and three dimensions. Two-dimensional visualizations are presented in the XZ projection.

**Figure 11.**A chromatic differentiation at various scale parameters in the polar and spherical coordinate systems (see on top and at the bottom of the figure, respectively). Visualizations in the polar coordinate system are presented in the XY projection.

**Figure 12.**Examples of visual displays (projections) with varying levels of contrast: (

**a**) a scale for converting the brightness of a point to its corresponding color; contrast ratios: (

**b**) 100%; (

**c**) 80%; (

**d**) 50%.

**Figure 13.**Chromatic differentiation renderings of Drosophila melanogaster chromosome X in the XY projection. Rectangular (

**a**), cyclic (

**b**), and polar (

**c**) coordinate systems.

**Figure 14.**Integral colored representations of chromosome 1 in Homo sapiens (

**a**), Morella rubra (

**b**), and Rattus norvegicus (

**c**) with scaling parameter of N = 300. One-dimensional representations (see three horizontal rows on the left), where the first line shows X, the second line shows Y, and the third line shows Z. Two-dimensional representations (see square displays on the right); the coordinates (top to bottom, abscissa/ordinate) are X/Y, Y/Z, and X/Z. Homo sapiens, visualizations of chromosome 1. Morella rubra, visualizations of chromosome 1. Rattus norvegicus, visualizations of chromosome 1.

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**MDPI and ACS Style**

Stepanyan, I.V.; Lednev, M.Y.
Chromatic Differentiation of Functional Mappings of the Composition of Nucleic Acids. *Symmetry* **2023**, *15*, 942.
https://doi.org/10.3390/sym15040942

**AMA Style**

Stepanyan IV, Lednev MY.
Chromatic Differentiation of Functional Mappings of the Composition of Nucleic Acids. *Symmetry*. 2023; 15(4):942.
https://doi.org/10.3390/sym15040942

**Chicago/Turabian Style**

Stepanyan, Ivan V., and Mihail Y. Lednev.
2023. "Chromatic Differentiation of Functional Mappings of the Composition of Nucleic Acids" *Symmetry* 15, no. 4: 942.
https://doi.org/10.3390/sym15040942