1 Overview

This is a R package to compute the automorphisms between pairwise aligned DNA sequences represented as elements from a Genomic Abelian group as described in reference (1). In a general scenario, whole chromosomes or genomic regions from a population (from any species or close related species) can be algebraically represented as a direct sum of cyclic groups or more specifically Abelian p-groups. Basically, we propose the representation of multiple sequence alignments (MSA) of length N as a finite Abelian group created by the direct sum of Abelian group of prime-power order:

\[ \qquad G = (\mathbb{Z}_{p^{\alpha_{1}}_1})^{n_1} \oplus (\mathbb{Z}_{p^{\alpha_{2}}_1})^{n_2} \oplus \dots \oplus (\mathbb{Z}_{p^{\alpha_{k}}_k})^{n_k} \]

Where, the \(p_i\)’s are prime numbers, \(\alpha_i \in \mathbb{N}\) and \(\mathbb{Z}_{p^{\alpha_{i}}_i}\) is the group of integer modulo \(p^{\alpha_{i}}_i\).

For the purpose of estimating the automorphism between two aligned DNA sequences, \(p^{\alpha_{i}}_i \in \{5, 2^6, 5^3 \}\).

1.1 Automorphisms

Herein, automorphisms are considered algebraic descriptions of mutational event observed in codon sequences represented on different Abelian groups. In particular, as described in references (3-4), for each representation of the codon set on a defined Abelian group there are 24 possible isomorphic Abelian groups. These Abelian groups can be labeled based on the DNA base-order used to generate them. The set of 24 Abelian groups can be described as a group isomorphic to the symmetric group of degree four (\(S_4\), see reference (4)).

For further support about the symmetric group on the 24 Abelian group of genetic-code cubes, users can also see Symmetric Group of the Genetic-CodeCubes., specifically the Mathematica notebook IntroductionToZ5GeneticCodeVectorSpace.nb and interact with it using Wolfram Player, freely available (for Windows and Linux OS) at, https://www.wolfram.com/player/.

2 Read the alignment FASTA and encode the sequences

A pairwise sequence alignment of protein coding regions SARS coronavirus GZ02 (GenBank: AY390556.1) and Bat SARS-like coronavirus isolate Rs7327 (GenBank: KY417151.1) is provided with the package.

3 Group representations

Group operations defined on the sets of DNA bases and codons are associated to physicochemical or/and biophysical relationships between DNA bases and between codons and aminoacids. In other words, a proper definition of a group operation on the set of bases or on the set of codons will encode the physicochemical or/and biophysical relationships between the set’s elements. Thus, by group operations defined on the set of bases or on the set of codons, we understand an encoding applied to represent specified physicochemical or/and biophysical relationships as group operations between the elements of the set. Then, we shall say that such an encoding permits the representation of DNA bases, codons, genes, and genomic sequences as elements from algebraic structures.

The DNA base set can be represented in 24 possible base orders, which leads to 24 possible representations of the genetic code. Each genetic code representation base-triplets on the Galois field GF(4) (or in GF(5)) leads to genetic code vector 3D-space, which is mathematically equivalent to a cube inserted in the 3D space (1). Each cube is denoted according to the corresponding base order.

Given a base-order, say ‘ACGT’, the Abelian group defined on this ordered set is isomorphic to the Abelian group defined on the set of integers modulo 4 (\(\mathbb{Z}_{4}\)). In practical terms, this is equivalent to replace each DNA base by the corresponding integer element. The base replacement in cube “ACGT and group”Z4" (\(\mathbb{Z}_{4}\)) is:

The base replacement in cube "ACGT and group ‘Z5’ (\(\mathbb{Z}_{5}\)):

After the DNA sequence is read, the corresponding codon sequences can be represented in the Abelian group \(\mathbb{Z}_{64}\) (i.e., the set of integers remainder modulo 64). The codon coordinates are requested on the cube ACGT. Following reference (4)), cubes are labeled based on the order of DNA bases used to define the sum operation.

The codon sequences (seq1 and seq2) with their corresponding coordinates (left) are returned, as well as the coordinated representation on \(\mathbb{Z}_{64}\) (coord1 and coord2).

3.1 “Dual” genetic-code cubes

The particular interest are the coordinate representation on “dual” genetic-code cubes. These are cubes where codons with complementary base pairs have the same coordinates in the corresponding cubes, as shown in reference (4)). Each pair of “dual” cubes integrates a group.

For example, let’s consider the complementary codons “ACG” and “TGC”, with complementary base pairs: A::T, C:::G, and G:::C, where symbol “:” denotes the hydrogen bonds between the bases.

Their representations on the dual cubes “ACGT” and “TGCA” on \(\mathbb{Z}_{4}\) are:

The sum of base coordinates modulo \(\mathbb{Z}_{4}\) is 3.

The same result for the same codon on different cubes

Their codon representation on \(\mathbb{Z}_{64}\) are:

The sum of base coordinates modulo \(\mathbb{Z}_{64}\) is 63.

The same result for the same codon on different cubes

4 Automorphisms on \(\mathbb{Z}_{64}\)

Automorphisms can be computed starting directly from the FASTA file. Notice that we can work only with genomic regions of our interest by giving the start and end alignment coordinates. In \(\mathbb{Z}_{64}\) automorphisms are described as functions \(f(x) = k\,x\quad mod\,64\), where \(k\) and \(x\) are elements from the set of integers modulo 64. Below, in function automorphism three important arguments are given values: group = “Z64”, cube = c(“ACGT”, “TGCA”), and cube_alt = c(“CATG”, “GTAC”).

In groups “Z64” and “Z125” not all the mutational events can be described as automorphisms from a given cube. The analysis of automorphisms is then accomplished in the set of dual genetic-code cubes. A character string denoting pairs of dual genetic-code cubes, is given as argument for cube. Setting for group specifies on which group the automorphisms will be computed. These groups can be: “Z5”, “Z64”, “Z125”, and “Z5^3”.

If automorphisms are not found in first set of dual cubes, then the algorithm search for automorphisms in a alternative set of dual cubes.

Observe that two new columns were added, the automorphism coefficient \(k\) (named as autm) and the genetic-code cube where the automorphism was found. By convention the DNA sequence is given for the positive strand. Since the dual cube of “ACGT” corresponds to the complementary base order TGCA, automorphisms described by the cube TGCA represent mutational events affecting the DNA negative strand (-).

The last result can be summarized by gene regions as follow:

That is, function automorphismByRanges permits the classification of the pairwise alignment of protein-coding sub-regions based on the mutational events observed on it quantitatively represented as automorphisms on genetic-code cubes.

Searching for automorphisms on \(\mathbb{Z}_{64}\) permits us a quantitative differentiation between mutational events at different codon positions from a given DNA protein-encoding region. As shown in reference (4) a set of different cubes can be applied to describe the best evolutionary aminoacid scale highly correlated with aminoacid physicochemical properties describing the observed evolutionary process in a given protein.

More information about this subject can be found in the supporting material from reference (4)) at GitHub GenomeAlgebra_SymmetricGroup, particularly by interacting with the Mathematica notebook Genetic-Code-Scales_of_Amino-Acids.nb.