Mutations of Nucleic Acids via Matroidal Structures

Abstract

The matroid concept is an important model in real life applications. Determining the existence of mutations of DNA and RNA plays an essential role in biological studies. The matroidal structures of matrices are used for determining the existence of mutations of DNA; graph theory and matroid theory can be used to identify important mutations in genetic data. We construct an algorithm to determine the existence of a mutation. Finally, we study the similarity and dissimilarity between genes using matroids.

1. Introduction

Matroidal structures have been an area of active research in mathematics for several decades which abstracts the notion of independence in linear algebra to more general settings. Matroids have applications in diverse fields such as computer science, combinatorics and optimization [1,2,3,4]. Matroidal structures are mathematical structures that capture the essential properties of independence and spanning in a variety of mathematical contexts including graph theory, geometry, combinatorics and algebra. Matroid theory was first introduced in the 1935s by Whitney [5] and has since become an important area of research in mathematics and computer science. Matroids can be found in many algebraic and combinatorial contexts. Both cycles in graphs and linear independence in vector spaces can be reduced to the same matroidal structure. Matroids are a subject of considerable interest due to their diverse fields, as noted in [6].

El Atik [7] describes the construction of new types of matroids called simplicial matroids using matrices and rough sets on simplicial complexes, explores the properties of these matroids including circuit and base axioms, rank functions and closure operators and provides examples of their applications.

El Atik [8,9] investigates topological structures in pre-approximation spaces including new lower types of pre-approximation spaces. Mutations can arise spontaneously, as a result of exposure to environmental factors such as radiation or carcinogens, or through errors in DNA replication or repair. The consequences of these mutations can range from single nucleotide changes to large-scale chromosomal aberrations and they can have significant effects on gene expression, protein function and cellular processes. Mutations in nucleic acids have been linked to a variety of diseases and disorders including cancer, genetic disorders and neurodegenerative diseases [10].

The omics technologies refer to a more comprehensive way of looking at the molecules that build up a cell, tissue and organism. The main role of genomics, transcriptomics and proteomics is to find all genes, messenger RNA (mRNA) and proteins in a specific biological sample without bias. It could also be used for something called ”high-dimensional biology”. The combination of all these techniques is known as systems biology. The fundamental aspect of these methods is a complicated system that, when looked at as a whole, is capable of eliciting a more in-depth comprehension than it would otherwise [11,12,13]. The application of computer technology to the management of biological data is bioinformatics. Gene-based medication discovery and development is aided using computers for data collection, storage, analysis and integration of biological and genetic data. The explosion of genomic information that has been made available to the public as a direct result of the genome project has prompted an increased need for capabilities in the field of bioinformatics.

We demonstrate the idea of a mutation space via an example involving genotypes [14]. Nucleotides are smaller molecules that combine to make DNA and RNA strands. These smaller molecules are called nucleotides. In DNA, the nucleotides are guanine (G), thymine (T) (or uracil (U) for RNA), adenine (A) and cytosine (C). The most important pairings for folding are guanine with cytosine and adenine with thymine (uracil). However, guanine and uracil can also pair. In theory, a nucleotide chain can fold and bond in many ways. Here, we use the nucleotide chain to illustrate the structure and the topological model from the graph; see [14,15,16,17]. Many topologists looked at topological models in biology [18,19] and medicine [20,21] to ascertain how DNA and RNA change from the point of view of multisets and topological structures.

Gioan [22] presents a study of complete graphs and their drawings with a focus on triangle mutations. The authors define triangle mutations as transformations of a complete graph drawing that preserve its planarity and connectivity while changing its embedding. They explore the properties of triangle mutations and provide examples of complete graph drawings that are equivalent to triangle mutations. Nieto et al. [23] suggest that a link between mathematics and the DNA structure may provide a better understanding of the DNA structure and its properties.

The study of mutations in nucleic acids is a fundamental area of research in genetics and molecular biology. Nucleic acids, such as DNA and RNA, carry the genetic information that determines the traits and characteristics of living organisms. Mutations in nucleic acids can have a variety of effects, ranging from benign to life-threatening and understanding the mechanisms behind these mutations is an important area of research.

This paper discusses the importance of the matroid concept in real-life applications and how it can be used to determine the existence of mutations in DNA and RNA. It also compares the validity of these mathematical results to the validity of these biological solutions. In particular, the use of matroidal structures in matrices is highlighted as a tool for identifying mutations, which is crucial in biological applications. This article starts off with a brief overview of the many notations and results of matroids. We approach DNA sequences as matrices and induce from them a matroid structure. The content continues to describe an algorithm that can be used to determine the existence of mutations. Finally, the content mentions that matroids can also be used to study the similarity and dissimilarity between genes. Determining the structure of matroids helps with solving important problems concerning DNA mutation in order to detect diseases and aid biologists in disease treatment.

2. Basic Concepts on Matroid Theory

2.1. Matroid Theory

Definition 1 ([6,24,25]).

A matroid structure M is an ordered pair $(E,T)$ composed of a finite set E that is known as a ground set and a collection T of subsets of E that meets the characteristics listed below:

($T_1$) $⌀\in T.$

($T_2$) If $A\in T$ and $B\subseteq A$, then $B\in T$.

($T_3$) $If A,B\in T$ and $\left|A\right|<\left|B\right|$ (where $\left|A\right|$ denotes the cardinality of A) then $\exists C\in T$ where $A\subseteq C\subseteq A\cup B.$

Remark 1.

In Definition 1, the condition $T_3$ can be written as: let $A,B\in T$ and $\left|A\right|\le \left|B\right|$ (or with $\left|B\right|=\left|A\right|+1)$, then $\exists a\in B-A$ such that $A\cup \left\{a\right\}\in T.$

The matriod M is typically represented by the notation $M=M(E,T)$. Each component of T is represented an indepentent set of M. Dependent sets are subsets of E that are not independent. Because of the criterion $T_1$, it is guaranteed that T is not empty; more specifically, it is guaranteed that it contains at least one subset of E. It is not possible to conclude this from $T_2$ or $T_3$, which indicates that each two maximally independent sets in matroid have the same cardinal number. As an illustration, consider the following examples.

Example 1.

If E = {a,b,c}, therefore $T=\{⌀,\left\{a\right\},\left\{b\right\},\left\{a,b\right\}\}$ is a matroid, but $T=\{⌀,\left\{a\right\},\left\{c\right\},\left\{a,b\right\}\}$ is not a matroid.

Definition 2 ([26]).

Let $M=\{E,T\}$. We have the following:

(i) 

The members of T are called the independent sets of M and symbolised by IND (M).

(ii) 

For any $K\subseteq ,E$ is said to be dependent if $K\notin T$ and is symbolised by D(M)

(iii) 

A set in T that is maximal in the sense of inclusion is called a base of the matroid M and is symbolised by B(M)

(iv) 

A minimal, in the sense of inclusion, dependent subset of E is called a circuit of the matroid M and is symbolised by C(M). The singleton circuit is called a loop. If {a,b} is a circuit, then a and b are said to be parallel.

(v) 

The rank function of the matroid is a function $f:P\left(E\right)\to N,definedby$ $f\left(B\right)=max\{\left|A\right|:A\subseteq B,A\in T\}$, for $B\subseteq E$.

(vi) 

For each $A\subseteq E$, the closure operator ${Cl}_M$ of a matrix M is defined as ${Cl}_M$(A) = $\{a\in E:f\left(a\right)=f(A\cup \left\{a\right\})\}$ and ${Cl}_M$(A) is called the closure of A in M. When there is confusion, we use the symbol $Cl\left(A\right)$. A is called a closed set if $Cl\left(A\right)=A.$

2.2. Matroid and Matrices

Definition 3 ([27]).

A basis of any subsets each $S\subseteq E$ of the columns of matrix A may be defined as a maximal linear independent set $J of S$. Maximal here means that there are no linear independent subsets of S that properly contain J.

Now, we present some examples to show that the independent set in matroids is an extension of the linearly independent set of vector spaces.

Example 2.

Assume that A is a matrix of which the columns are indexed by $E=\{a,b,c,d\}$. $A=\begin{array}{}\begin{array}{cccc}a& b& c& d\end{array}\\ \left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right)\end{array},$ $T=\{⌀,\left\{a\right\},\left\{b\right\},\left\{c\right\},\left\{d\right\},\left\{a,b\right\},\{a,c\},\left\{a,d\right\},\left\{b,c\right\},\{b,d\},\left\{c,d\right\},\left\{b,c,d\right\},\{a,b,c\}\}$ and a circuit $C=\left\{\{a,b,d\}\right\}.$

Example 3.

Suppose a $3\times 5$ matrix of which the columns are indexed by $E=\{1,2,3,4,5\}$. Consider linear independence among column vectors. $A=\begin{array}{}\begin{array}{ccccc}1& 2& 3& 4& 5\end{array}\\ \left(\begin{array}{ccccc}1& 0& 0& 1& 0\\ 0& 1& 0& 1& 1\\ 0& 1& 1& 0& 1\end{array}\right)\end{array},$ Then: T = $\{⌀,\left\{1\right\},\left\{2\right\},\left\{3\right\},\left\{4\right\},\left\{5\right\},\{1,2\},\left\{1,3\right\},\left\{1,4\right\},\left\{1,5\right\},\left\{2,3\right\},\left\{2,4\right\},\left\{2,5\right\},\left\{3,4\right\},\{3,5\},\left\{4,5\right\},\left\{1,2,3\right\},\{1,2,5\},\left\{1,3,4\right\},\left\{1,3,5\right\},\{1,4,5\},$$\left\{2,3,4\right\},\left\{2,4,5\right\},\left\{3,4,5\right\}\}$ is a matroid. For A = {1, 2} and B = {2, 3, 4}, B-A = {3, 4}, for instance, we can take a = 3 to obtain $A\cup \left\{3\right\}=\{1,2,3\}\in T$, whereas a = 4 leads to $A\cup \left\{4\right\}=\{1,2,4\}\notin T$.

The family β of the maximal members of T is given by $\beta =$ $\{\left\{1,2,3\right\},\{1,2,5\},\left\{1,3,4\right\},\left\{1,3,5\right\},\{1,4,5\},\left\{2,3,4\right\},\left\{2,4,5\right\},\left\{3,4,5\right\}\}$ in which the members of T and only of the maximal members of T (maximal with the respect inclusion) are called a base of M. Note that the family β satisfies: for $B_1$, $B_2$$\in \beta$ and for $x\in B_1-B_2,\exists y\in B_2-B_1$ where ($B_1-\left\{x\right\}$)$\cup (B_2-\left\{y\right\})\in \beta$. For $B_1=\left\{1,2,3\right\}$, $B_1=\{3,4,5\}$, $B_1-B_2=\left\{1,2\right\},B_2-B_1=\left\{4,5\right\}.ifx=1forexample,wetakey$ = 4 to obtain ($B_1-\left\{x\right\}$)$\cup \left(B_2-\left\{y\right\}\right)=\{2,3,4\}\in \beta$. Additionally, the circuit is $C=\left\{2,3,5\right\}.$

Example 4.

Let matrix $A=\begin{array}{}\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\end{array}\\ \left(\begin{array}{cccccc}0& 0& 10& 1& 0& 0\\ 0& 1& 1& 0& 0& 1& 1\\ 1& 0& 0& 0& 0& 1& 1\end{array}\right)\end{array},$ Then one can deduce that T = $\{⌀,\left\{1\right\},\left\{2\right\},\left\{3\right\},\left\{5\right\},\left\{6\right\},\left\{7\right\},\{1,2\},\left\{1,3\right\}$, $\left\{1,5\right\},\left\{1,6\right\},\left\{1,7\right\},\left\{2,3\right\},\left\{2,5\right\},\left\{2,6\right\},\left\{2,7\right\},\left\{3,5\right\},\left\{3,6\right\},\{3,7\}$, $\left\{5,6\right\},\left\{5,7\right\},\left\{1,2,3\right\},\left\{1,2,5\right\},\left\{1,3,5\right\},\left\{1,3,6\right\},\left\{1,3,7\right\},\left\{1,5,6\right\},\left\{1,5,7\right\},\left\{2,3,6\right\},\left\{2,3,7\right\},\left\{2,5,6\right\},\left\{2,5,6\right\},\left\{2,5,7\right\},\left\{3,5,6\right\},\{3,5,7\}\}$ is a matroid. The family of circuit is C = {{4}, {6, 7}, {1, 2, 6}, {1, 2, 7}, {2, 3, 5}, {1, 2, 5, 6}, {1, 3, 5, 7}}.

In the following example, we can simply write the matroid and its circuit in other forms.

Example 5.

Let matrix $A=\begin{array}{}\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\end{array}\\ \left(\begin{array}{ccccccc}1& 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 1& 1& 1& 0\\ 0& 0& 1& 0& 1& 1& 0\end{array}\right)\end{array},$ Then: $T=P\left(E\right)-\left\{7\right\}\mathrm{is}a\mathrm{matroid}$ and $C=\left\{\left\{2,2,4\right\},\left\{2,3,5\right\},\left\{2,3,6\right\}\right\}$ and any subsets containing {5, 6}.

The matrix can also define a field which is similar to vector spaces.

2.3. Matroids and Graph Theory

Graph G can be thought of as a zero-one matrix A, mod 2, where each column contains exactly two ones. The rows of A represent the nodes and the columns represent the edges in a graph. If there is one in a given row and column, then there is a connection between that node and the adjacent edge. Given that the graph is connected, which means that the row of A cannot be partitioned into two non-empty sets in such a way that every column has both of its ones in the same set, the column bases of A are identical to the edge sets of the spanning-trees of G and the linearly independent sets of columns are identical to the edge sets of forests in G. To define a matroid from a graph, we will set the ground set E to be the set of edges. Then the independent sets will be those sets of edges that do not contain a cycle [4]. By Definitions 1, 2 and basic definitions in graph theory, one can easily deduce Proposition 1.

Proposition 1.

(i) 

$A\in T$ if A does not contain a cycle of G.

(ii) 

B is a circuit of $M\left(G\right)$ if B is a cycle of G.

(iii) 

B is a base of $M\left(G\right)$ if B is a spanning forest of G.

The following two examples are of a graphic matroid.

Example 6.

Suppose that G is a graph as shown in Figure 1, then the edge set $E=\{a_1,a_2,a_3,a_4\}$ from the ground set of a matroid whose independent sets are T = $\{⌀,\left\{a_1\right\},\left\{a_2\right\},\left\{a_3\right\},\left\{a_4\right\},\left\{a_1,a_2\right\},\{a_1,a_3\},\left\{a_1,a_4\right\},\left\{a_2,a_3\right\}$,$\left\{a_2,a_4\right\},\left\{a_3,a_4\right\},\left\{a_1,a_2,a_4\right\},\left\{a_2,a_3,a_4\right\}\},$ and the base will be $\beta \left(M\right)=max\left(T\right)=\{\left\{a_1,a_2\right\},\left\{a_1,a_3\right\},\left\{a_1,a_4\right\},\left\{a_2,a_3\right\},\{a_2,a_3,a_4\}\}$. The dependent set $D\left(M\right)=\left\{\{a_1,a_2,a_3\}\right\}$. Hence, $C\left(M\right)=D\left(M\right).$

Example 7.

Assume that G is a graph as shown in Figure 2, then the edges $E=\{a_1,a_2,a_3,a_4,a_5\}$ from the ground set of a matroid whose independent sets are $T=\{⌀,\left\{a_1\right\},\left\{a_2\right\},\left\{a_3\right\},\left\{a_4\right\},\left\{a_5\right\},\left\{a_1,a_2\right\},\left\{a_1,a_3\right\},\left\{a_1,a_4\right\},\left\{a_1,a_5\right\},\left\{a_2,a_3\right\},\left\{a_2,a_4\right\},\left\{a_2,a_5\right\},\left\{a_3,a_5\right\},\left\{a_4,a_5\right\},\left\{a_1,a_2,a_3\right\},\left\{a_1,a_2,a_4\right\},\left\{a_1,a_2,a_5\right\},$$\left\{a_1,a_3,a_5\right\},\left\{a_1,a_4,a_5\right\},\left\{a_2,a_3,a_5\right\},\left\{a_2,a_4,a_5\right\}$} and $C,$ a set of circuits of $M\left[G\right]$ is $,\left\{a_3,a_4\right\},\left\{a_1,a_2,a_4,a_5\right\},\left\{a_1,a_2,a_3,a_5\right\}$ and the base will be $\beta \left(M\right)=max\left(T\right)=\{\left\{a_1,a_3\right\},\left\{a_1,a_4\right\},\left\{a_1,a_5\right\},\left\{a_2,a_3\right\},\left\{a_2,a_4\right\},\left\{a_2,a_5\right\},\{a_1,a_2,a_5\}\}$. The dependent sets are $D\left(M\right)=\left\{\left\{a_3,a_4\right\},\right.\left.\left\{a_1,a_2,a_3,a_5\right\},\{a_1,a_2,a_4,a_5\}\right\}$. It is clear that $C\left(M\right)=D\left(M\right).$

Definition 4 ([6]).

Let $M_1,M_2$ be two isomorphic matroid structures if there exist a bijection $\psi :E\left(M_1\right)\to E\left(M_2\right)$ since $\forall A\in E\left(M_1\right)$, $\psi \left(A\right) isanindependent$ set of $M_2$ if A is an independent set in $M_1.$

Remark 2.

By comparing the dependent sets which are $M\left[G\right]$ with $M\left[A\right]$ in Example 7, we see that, under the bijection $\psi :E\left(G\right)\to E\left(A\right)$ is defined as $\psi \left(a_i\right)$ $=i$, a set $A$ is a circuit in $M\left[G\right]$ if $\psi (A$) is a circuit in $M\left[A\right]$. In the same way, a set B is an independent set in $M\left[G\right]$ if $\psi (B$) is an independent set in $M\left[A\right].$

Example 8.

Consider G be a graph as shown in Figure 3. Then the T = $\{⌀,\left\{a_1\right\},\left\{a_2\right\},\left\{a_3\right\},\left\{a_4\right\},\left\{a_1,a_3\right\},\{a_1,a_4\},\left\{a_2,a_3\right\}\left\{a_2,a_4\right\}$,$\left\{a_3,a_4\right\}\},$ and the base will be $\beta \left(M\right)=max\left(T\right)=\{\left\{a_1,a_3\right\},\left\{a_1,a_4\right\},\left\{a_2,a_3\right\},\left\{a_2,a_4\right\},\left\{a_3,a_4\right\}\}$. The dependent sets are $D\left(M\right)=\{\left\{a_5\right\},\left\{a_1,a_5\right\},\left\{a_1,a_4\right\},\left\{a_3,a_5\right\},\left\{a_4,a_5\right\},\left\{a_1,a_2\right\},\left\{a_1,a_3,a_4\right\},\left\{a_2,a_3,a_4\right\}\}$. It is obvious that $C\left(M\right)=maxD\left(M\right)=\left\{\left\{a_1,a_2\right\},\left\{a_1,a_3,a_4\right\},\left\{a_2,a_4,a_4\right\},\left\{a_5\right\}\right\}\ne D\left(M\right).$ In this case, if G has a loop, then $C\left(M\right)\ne D\left(M\right).$

Proposition 2.

In a matroid structure $M=(E,T)$, the following statements hold:

(i) 

$A\in T$ if and only if $f\left(A\right)=\left|A\right|$, equivalently, $A\in D\left(M\right)$ if and only if $f\left(A\right)\le \left|A\right|$.

(ii) 

$f\left(A\right)=f(A\cup B)$ if and only if $B\subseteq Cl\left(A\right),\forall B\subseteq E.$

Proof. 

According to Definition 1 and Remark 1, the proof is straightforward. □

3. Matroidal Structure Induced by Topological Operators

We define a matroidal structure in terms of topological operators in this section.

Let X be a non-empty finite set and let F be a family of subsets of $X,$ that is $F\subseteq P\left(X\right)$.

Definition 5 ([28]).

Let a family F on X be a preclosure system (PCS) when it satisfies these conditions:

(i) 

$X\in F$.

(ii) 

$If{F}_1,F_2\in F,then{F}_1\cap F_2\in F$.

Proposition 3.

Let F be a preclosure system. If $F_1\in F$, $F_1$ is preclosed. Let ${pCl}_F\left(A\right)=\cap \left\{F_1\in F:\right.\left.A\subseteq F_1\right\},$ then it is simple to verify that for all $A,B\subseteq X,$ the next properties hold:

(PCL1) 

$A\subseteq {pCl}_F\left(A\right);$

(PCL2) 

If $A\subseteq B$, then ${pCl}_F\left(A\right)\subseteq {pCl}_F\left(B\right)$;

(PCL3) 

${pCl}_F\left({pCl}_F\left(A\right)\right)={pCl}_F\left(A\right);$

(PCL4) 

${pCl}_F\left(⌀\right)=⌀$;

(PCL5) 

$For anyA,B\subseteq X,{pCl}_F\left(A\cup B\right)={pCl}_F\left(A\right)\cup {pCl}_F\left(B\right)$;

(PCL6) 

$For anyA\subseteq X,{x\in X,ify\in pCl}_F\left(A\cup \left\{x\right\}\right)-{pCl}_F\left(A\right),thenx\in {pCl}_F\left(A\cup \left\{y\right\};\right)$

(PCL7) 

For any $A\subseteq X,{pCl}_F\left({\left[{pCl}_F\left(A\right)\right]}^c\right)={\left[{pCl}_F\left(A\right)\right]}^c.$

Proof. 

The properties from PCL1 to PCL5 are obvious. It is sufficient to prove (PCL6). Since ${y\in pCl}_F\left(A\cup \left\{x\right\}\right)-{pCl}_F\left(A\right),\mathrm{then}$ ${y\in pCl}_F\left(A)or{y\in pCl}_F(\left\{x\right\}\right)$ and $y\notin {pCl}_F\left(A\right)$ then ${y\in pCl}_F(\left\{x\right\}$. So $x\in {pCl}_F\left(\left\{y\right\}\right)$ and so $x\in {pCl}_F\left(A\cup \left\{y\right\}\right).$ PCL7 is directly proven by PCL6. □

Definition 6.

A map $pCl:P\left(X\right)\to P\left(X\right)$ is named a preclosure operator (PCO) on X if pCl satisfies the above conditions (PCL1), (PCL2) and (PCL3).

(i) 

By Definition 5, it is simple to verify this. $F_{\mathrm{pCl}}$ = $\{A\subseteq X:pCl\left(A\right)=A\}$ is a preclosure system on X.

(ii) 

A preclosure operator that satisfies (PCL4) and (PCL5) is called a Kuratowski preclosure operator (KPO), which determines a supra topology on X.

(iii) 

A matroid structure is defined by a preclosure operator that satisfies (PCL6), which we refer to as a matroidal preclosure operator (MPO) and is defined by $T_{pCl}=\{A\subseteq X:\forall x\in A,x\notin pCl(A-\left\{x\right\})\}.$

Proposition 4.

The structure ($X,T_{\mathrm{pCl}}$) is a matroid.

Proof. 

(i)

That is obvious $⌀\in T_{\mathrm{pCl}}.$

(ii)

If $A\in T_{\mathrm{pCl}}$ and $B\subseteq A$ and for every $x\in A,$ then $x\notin pCl\left(A-\left\{x\right\}\right).$ Therefore, there exists a preopen set G where $G\cap \left(A-\left\{x\right\}\right)=⌀$ and so $G\cap A=⌀$. By $B\subseteq A$, since $G\cap B=⌀$ and $G\cap \left(B-\left\{y\right\}\right)=⌀$ for $y\in B.$ Therefore, $y\notin pCl\left(B-\left\{y\right\}\right)$ and $B\in T_{\mathrm{pCl}}.$

(iii)

By the fact that $pCl\left(A\right)\subseteq pCl\left(A\right)$ for any subset A, we have that if $A,B\in T_{\mathrm{pCl}}$ since $\left|A\right|\le \left|B\right|$, therefore there exists $x\in B-A$ where $A\cup \left\{x\right\}\in T_{\mathrm{pCl}}$.

□

Rough set theory states that a preclosure operator pCl is an upper approximation operator (UPO) if pCl satisfies (PCL5) and (PCl7).

Let $KPS,$ MPS and UPS be the preclosure system corresponding to the preclosure operator $KPO,MPO\mathrm{and}UPO,$ respectively. We will now discuss the relation between these three types of preclosure operators (system).

Theorem 1.

A UPO is a $KPOandMPO$.

Proof. 

Consider $\mathrm{pCl}\mathrm{is}\mathrm{a}\mathrm{UPO}.$ As is known, a $\mathrm{UPO}\mathrm{is}\mathrm{a}\mathrm{KPO}$, so we simply prove that pCl is a MPO. For $A\subseteq X$, $x\in X$ and $y\in pCl\left(A-\left\{x\right\}\right)-pCl\left(A\right)$ by (PCL5), $y\in pCl\left(\left\{x\right\}\right)$. To prove $x\in pCl\left(A-\left\{y\right\}\right)$, we need merely to show $x\in pCl\left(\left\{y\right\}\right)$, by (PCL2). If $x\in {\left(pCl\left(y\right)\right)}^c$, then $\mathrm{pCl}\left(\left\{x\right\}\right)\subseteq pCl({\left(pCl\left(\left\{y\right\}\right)\right)}^c$)= $({\left(\mathrm{pCl}\left(\left\{y\right\}\right)\right)}^c$. That is $\mathrm{pCl}\left(\left\{y\right\}\right)\subseteq ({\left(pCl\left(\left\{x\right\}\right)\right)}^c$. It follows (PCL3) and (PCL7) that $\mathrm{pCl}\left(\left\{y\right\}\right)=pCl\left(pCl\left(\left\{y\right\}\right)\right)\subseteq pCl\left(\right(pCl{\left(\left\{x\right\}\right))}^c$) = ${\left(pCl\left(\left\{x\right\}\right)\right)}^c,$ this contradicts with $y\in pCl\left(\left\{x\right\}\right).$ Hence, (PCL6) holds and $\mathrm{pCl}\mathrm{is}\mathrm{a}$ MPO. □

Remark 3.

The following is the preclosure systems diagram that corresponds to Figure 4, which is as follows:

Proposition 5.

IF $(V,D)$ is a MPS and KPS, then $(V,D)$ is a UPS.

Proof. 

Assuming that $pc{l}_D$ is the preclosure operator caused by (V, J), we will show that $pc{l}_D$ is a UPO. To prove this, we just need to prove a partition $V/R=V_1,V_2,\dots ,V_n$ on V since $pc{l}_D\left(S\right)=\bigcup \{V_i\in V/R|V_i\cap S\ne \varphi \}$($\forall S\subseteq U)$. Such that $pc{l}_D$ is a KPO, $pc{l}_D\left(\varphi \right)=\varphi$. That is, $\varphi \in D$. $(V,D)$ is a MPS, so D includes the following: $\forall J\in D$, if $J_1,J_2,\dots ,J_k$ is the family of preclosed sets that cover J, therefore $J_1-J,J_2-J,\dots ,J_k–J$ partition $V-J$. Suppose that $D^{*}=\{J_{i1},J_{i2},\dots ,J_{im}\}$ is the family of preclosed sets that cover $\varphi$, then $D^{*}$ is a partition of V. $\forall J\in D$, we show that J is the union of some elements of $D^{*}$. $\forall s\in J$, where $D^{*}$ is a partition of V, there is $J_s\in D^{*}$ since $s\in J_s$. We claim that $J_s\subseteq J$, or else, $J_s\cap J\in D$ and $J_s\cap J\subset J_s$, contradicts that $J_s$ is a preclosed set covering $\varphi$. Hence we have $J=\bigcup \{J_s:s\in J\}$. Now we show $pc{l}_D\left(X\right)=\bigcap \{J\in D:S\subseteq J\}=\bigcup \{J_{ij}\in D^{*}:J_{ij}\cap S\ne \varphi \}$($\forall S\subseteq U)$.

Let $s_0\in \bigcap \{J\in D:S\subseteq J\}$, we let $J_{s_0}\cap S=\varphi$ and find a contradiction. $\forall J\in D$ and $S\subseteq J$, following from $J=\bigcup \{J_s:s\in J\}$ that $J–J_{s_0}\in D$ and $S\subseteq J-J_{s_0}$. therefore $J_{s_0}\cap (\bigcap \{J\in D:S\subseteq J\})=\varphi$, in contrast to $s_0\in \bigcap \{J\in D:S\subseteq J\}$. Then $J_{s_0}\cap S\ne \varphi$ and $s_0\in \bigcup \{J_{ij}\in D^{*}:J_{ij}\cap S\ne \varphi \}$. That is $\bigcap \{J\in D:S\subseteq J\}\subseteq \bigcup \{J_{ij}\in D^{*}:J_{ij}\cap S\ne \varphi \}$.

We now point in a new direction. Let $q_0\in \bigcup \{J_{ij}\in D^{*}:J_{ij}\cap S\ne \varphi \}$. Since $D^{*}$ is a partition of U and $q_0\in J_{q_0},J_{q_0}\cap S\ne \varphi$. For each $J\in D$ satisfying $S\cap J$, it follows $J=\bigcup J_s:s\in J$ that J contains $J_{q_0}$. Thus, J contains $q_0$, and $q_0\in \bigcap \{J\in D:S\subseteq J\}$. □

Graph theory is a mathematical framework used to study the properties and relationships of objects represented as nodes or vertices connected by edges. In the context of genetics, graph theory can be applied to analyze the relationships between genes and their mutations. Using graph theory, researchers can analyze the properties of this network of genes and mutations, and such analysis can help identify key mutations that play important roles in disease development and progression. Graph theory and matroid theory are two mathematical frameworks that can be used to analyze the structure of mutations and their relationships in genetic data. Both graph theory and matroid theory can be used to identify important mutations in genetic data. Additionally, these frameworks can be used to identify mutations, which can provide insights into the molecular mechanisms underlying disease.

4. Mutations via Their Graph and Matroidal Structures

In this section, we study a substitution mutation, an insertion mutation and a deletion mutation with a new procedure using matroids with graphs.

DNA Structure and Mutations

Figure 5 depicts the transcription process in which an RNA polymerase creates an RNA copy from a section of DNA. Translation is the process by which polypeptide sequences are created using the translated mRNA as a template. The type of the RNA’s coding that is actually read during the production of polypeptides is messenger RNA (mRNA), which is coded in stacks of RNA [29,30]. The majority of genes are now discovered at the DNA level before they are discovered as mRNA or as a portion product. The fundamental tenet of molecular biology is the explanation of how genetic information is transferred from one component of a biological system to another. It is well known that DNA leads to RNA and RNA, in turn, leads to protein. Every C is linked with G and vice versa. On the other side, A is linked with $T\equiv U$ and vice versa. Otherwise, a mutation will occur. There are different types of mutations including substitution, insertion, deletion.

(i)

A mutation that exchanges one base for another is called substitution as in Figure 6.

(ii)

Insertion mutation occurs when extra base pairs are inserted as in Figure 7.

(iii)

If a section of DNA is lost or deleted, then the mutation is called a deletion as in Figure 8.

Matroids have many applications in computer science, combinatorial optimization and algorithm design. For example, matroids are used in graph theory to model the structure of graphs and in optimization theory to model various optimization problems, such as the minimum spanning tree problem and the maximum flow problem. Matroids are also used in coding theory to construct error-correcting codes and in machine learning to model complex data structures [31,32]. In graph theory [16], the set of vertices will be denoted by V of a finite set. The set of edges had the form $E\left(V\right)=$$\left\{\right\{u,v\}$ s. t. $u,v\in V$$u\ne v\}$. In other words, $u,v$ are called adjacent vertices.

Recent studies have determined the existence of the mutation by the distance function, relations and topology. In the following, we will use graph theory to determine the mutation of genes which do not require a certain length.

One of the challenges in genetic research is the handling of gene sequences, including converting these sequences into numerical data, identifying relationships between them and organizing them in tables, as in Table 1 (analysis); for more information, see [19]. In this regard, the application of graph and matrix theory provides a promising approach for analyzing genetic data. By representing genes as graphs and matrices, researchers can more effectively visualize and analyze genetic information, potentially leading to new insights into the fundamental nature of genetic structures.

Definition 7.

A graph $G(V,E)$ on genes is defined as a set of nucleotide $\{A,C,G,T\}$ from the sequence of DNA. In other words, it is the set of all vertices V and the edges $x,y$ between the vertices such that $\mathcal{R}\left(x\right)\cap \mathcal{R}\left(y\right)\ne \varphi$ such that $x,y\in V$, where $\mathcal{R}\left(x\right)$ will denote the set of vertices incident with x.

In the following, we discuss the existence of mutations via graph and matroidal structures. This can be considered throughout the discussion of the following examples and results.

Example 9.

Let $\mathcal{R}=$$\left\{\right(C,G),(T,A),$$(G,C),(A,T)\}$. Then, the induced graph through the relation $\mathcal{R}$ is

Proposition 6.

Let the types be strings of bits, vectors and DNA or RNA sequences such that a mutation has not occurred. Then, its graph structure will be undirected.

Proof. 

Consider the types that do not have a mutation. Then the relation between types (${\mathcal{M}}_1$ and ${\mathcal{M}}_2$) will be only a symmetric relation. So, the type of graph will be undirected. □

Example 10.

Let $\mathcal{R}=$$\left\{\right(T,A),(C,A),$$(G,A),(A,T),(A,C),$$(G,C),(C,G)\}$. Then its graph structure is

Corollary 1.

If the types contain a mutation, then its graph structure is directed.

Corollary 2.

If the types do not contain a mutation, then its graph structure in terms of multiset relation is directed.

Now, we state an Algorithm 1 which determines whether a DNA sequence has a mutation by using a graph representation of the sequence. It examines each edge in the graph and connects it with non-cyclic edges. It then constructs a matroid from the resulting graph to represent the independence structure of the edge set. Algorithm 1 checks if any subset of the edge set belongs to the matroid. If it does, there is no mutation and if it does not, there is a mutation. However, Algorithm 1 assumes that the input graph is a valid representation of the DNA sequence and that the mutation alters the graph structure.

Algorithm 1: Mutation via matroids and graphs.

Input: A graph G = (V,E) from DNA stand.    Output: The existence of mutation in DNA or not   1: for (ei ∈ E(G))   2: connect ei with edges ej that have not cycle.   3: Construct the matroid $M_{\mathrm{DNA}}$.   4: if A ∈ P (E(G))   5: A ∈$M_{\mathrm{DNA}}$.   6: Then, DNA has a mutation   7: Otherwise, DNA has no mutation   8: end if   9: end for

Example 11.

Arabidopsis Thaliana Gamma-Glutamylcysteine Synthetase Gene (abbreviated as CAD2) [33]

Tair Accession:	1005028114.
GenBank Accession:	AF068299.
Sequence Length	5277.

$5^{\prime }$  AT CGATATGTAACACAAT ⋯ TGTATGTTTTT  $3^{\prime }$;

$3^{\prime }TAGCTATACATTGTGTTA\cdots ACATACAAAAA{5}^{\prime }$;

By using MSC-code [19], we obtain the data in Table 1.

Table 1. Bonding between nucleotide.

	A	T	C	G
A	0	1859	0	0
T	1543	0	0	0
C	0	0	0	1019
G	0	0	856	0

Table 1. Bonding between nucleotide.

	A	T	C	G
A	0	1859	0	0
T	1543	0	0	0
C	0	0	0	1019
G	0	0	856	0

Then its graph structure is

By Algorithm 1 it is evident that there is no mutation.

Example 12.

If we locate a mutation in CAD2 [33], then by MSC-code, we obtain the data in

Table 2. Bonding between nucleotides.

	A	T	C	G
A	203	1351	175	130
T	1091	154	124	171
C	130	202	47	633
G	118	149	510	78

.

The graph structure induced by Table 2:

Since $e_1=1091$, $e_2=1351$, $e_3=633$, $e_4=510$, $e_5=154$, $e_6=203$, $e_7=47$, $e_{5}8=78$, $e_9=149$, $e_{10}=171$, $e_{11}=124$, $e_{12}=202$, $e_{13}=130$, $e_{14}=175$, $e_{15}=118$, $e_{16}=130$. By Algorithm 1, there exists a mutation. The number of mutations can be calculated with $e_5+e_6+\cdots +e_{16}=1681$—the same result as obtained from code MSC-code [19].

We would like to clarify that our focus in this section was to introduce a novel approach to modelling the mutations of nucleic acids using matroidal structures. While the basic concepts of matroidal structures may be known to some mathematicians, we believe that the application of this theory to the study of nucleic acid mutations is a novel contribution to the field, we also acknowledge that simply naming mathematical expressions does not produce new knowledge. However, our work goes beyond a mere description of matroidal structures and demonstrates their potential application to the study of nucleic acid mutations, we provide examples of how matroidal structures can be used to model the behavior of nucleic acids and predict the occurrence of mutations. We believe that these applications demonstrate the potential value of matroidal structures in the context of nucleic acid research. We also believe that this section provides a valuable contribution to the field by introducing a new approach to modelling the mutations of nucleic acids. By using matroidal structures, we are able to capture the underlying combinatorial structure of nucleic acids and provide insights into their behavior.

5. Matroidal Structure of DNA via Matrices

Previous studies [18,19,34,35] to identify mutations depend on establishing a function and proving that it is a measurement function and require many calculations. The matroid method is easier and depends on the bonding between the nucleotides. In this section, we study a mutation with new procedure using matroids with a matrix. We created Algorithm 1 to determine the existence (or absence) of a mutation. Algorithm 1 indicates how many (A, T, G and C) elements there are; through Algorithm 1, the gene sequence was converted into a matrix.

Definition 8.

Let ${\mathcal{M}}_{5^{\prime }}^{3^{\prime }}\equiv 5^{\prime }\dots \dots 3^{\prime }$ be a sense strand of DNA (the first tape in wild type) and ${\mathcal{M}}_{3^{{}^{\prime }}}^{5^{\prime }}\equiv 3^{\prime }\dots \dots 5^{\prime }$ be an antisense strand of DNA (the second tape in wild type).

As shown in Figure 9, after finding the relationship between the gene sequence by [19] and converting it into a table as in Table 1 and Table 2, the table can be converted into a matrix by Algorithm 2.

Algorithm 2: Mutation via matroids and matrices.

Input: A matrix (aij) from DNA stand, where i indicates to row and j indicates to column.      Output: the existence of mutation in DNA or not.    1: for nucleotides {A,T,C,G} ∈ DNA stand.    2: if {A,T,C,G} are connect    3: Then, aij = 1    4: else    5: Then, aij = 0    6: end if    7: end for    8: for nucleotides {A,T,C,G} ∈ DNA stand    9: if all columns (aij) are independent vectors    10: Then, DNA has not mutation    11: Otherwise, DNA has mutation    12: end if    13: end for

Example 13.

(i) 

Consider the following DNA strand, $5^{\prime }\dots CTGCAG\dots 3^{\prime }$ and $3^{\prime }\dots GACGTC\dots 5^{\prime }$. By Algorithm 2, the matrix $M_1$$=\begin{array}{cc}& \begin{array}{cccc}A& T& C& G\end{array}\\ \begin{array}{c}A\\ T\\ C\\ G\end{array}& \begin{array}{c}\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)\end{array}\end{array}$, where the row $\left(ATCG\right)$ represents the first tape in wild type $M_{5^{\prime }}^{3^{\prime }}$ and the column ${\left(ATCG\right)}^t$ represents the second tape in wild type $M_{3^{\prime }}^{5^{\prime }}$, if $X=\{A,T,C,G\}$, the structure for the matrix $M_1 is{M}_{\mathrm{DNA}}=\{\varphi ,\left\{A\right\},\left\{T\right\},\left\{C\right\},\left\{G\right\},\{A,T\},\{A,C\},\{A,G\},\{T,C\},\{T,G\},\{C,G\},\{A,T,C\},\{A,T,G\},\{A,C,G\},\{T,C,G\},\{A,T,C,G\}\}.$ We observe that $\beta \left(M_{\mathrm{DNA}}\right)=\{\{A,T,C\},\{A,T,G\},\{T,C,G\}\}$ which means that all vectors of matroid $M_{\mathrm{DNA}}$ are independent. Then there is no mutation in this DNA sequence.

(ii) 

Consider the following DNA strand, $5^{\prime }\dots ACTAG\dots 3^{\prime }$ and $3^{\prime }\dots CTAGA\dots 5^{\prime }$ By Algorithm 2, the matrix $M_1=\begin{array}{cc}& \begin{array}{cccc}\begin{array}{c}A\end{array}& \begin{array}{c}T\end{array}& \begin{array}{c}C\end{array}& \begin{array}{c}G\end{array}\end{array}\\ \begin{array}{c}A\\ T\\ C\\ G\end{array}& \begin{array}{c}\left(\begin{array}{cccc}0& 1& 0& \begin{array}{|c|}\hline 1\\ \hline\end{array}\\ 0& 0& \begin{array}{|c|}\hline 1\\ \hline\end{array}& 0\\ \begin{array}{|c|}\hline 1\\ \hline\end{array}& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right)\end{array}\end{array},$ if $X=\{A,T,C,G\},$ the structure for the matrix M₂ is $M_{\mathrm{DNA}}$ = $\{\varphi ,\{A\},\{T\},\{C\},\{G\},\{A,T\},\{A,C\},\{A,G\},\{T,C\},\{C,G\},\{A,T,C\},\{A,C,G\left\}\right\}$. We observe that C($M_{\mathrm{DNA}}$) = $\left\{\right\{T,G\},\{A,T,G\},\{T,C,G\},\{A,T,C,G\left\}\right\}$ which means that not all vectors of the matroid $M_{\mathrm{DNA}}$ are independent. There is substitution mutation in this DNA strand.

Example 14.

Continue for Example 11;

By MC-code [19] and Algorithm 2, we get the

Table 3. Bonding between nucleotide.

	A	T	C	G
A	0	1859	0	0
T	1543	0	0	0
C	0	0	0	1019
G	0	0	856	0

and

Table 4. Relation between nucleotides.

	A	T	C	G
A	0	1	0	0
T	1	0	0	0
C	0	0	0	1
G	0	0	1	0

.

Then the matrix $M_1=\begin{array}{cc}& \begin{array}{cccc}A& T& C& G\end{array}\\ \begin{array}{c}A\\ T\\ C\\ G\end{array}& \begin{array}{c}\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)\end{array}\end{array}$, if $X=\{A,T,C,G\}$, the structure for the matrix $M_{1}is{M}_{\mathrm{DNA}}=\{\varphi ,\left\{A\right\},\left\{T\right\},\left\{C\right\},\left\{G\right\},\{A,T\},\{A,C\},\{A,G\},\{T,C\},\{T,G\},\{C,G\},\{A,T,C\},\{A,T,G\},\{A,C,G\},\{T,C,G\},\{A,T,C,G\}\}\}.$ We observe that $\beta \left(M_{\mathrm{DNA}}\right)=\{\{A,T,C\},\{A,T,G\},\{T,C,G\}\}$ which means that all vectors of matroid $M_{\mathrm{DNA}}$ are independent. There is no mutation in this DNA sequence. These results correspond to the National Center for Biotechnology Information (NCBI) [33].

6. A Similarity and Dissimilarity between the Sequences of DNA

We can define a similarity and dissimilarity in matrorid M = M(E, T) as

A similarity =$\frac{∥T∥}{∥P\left(E\right)∥}$;

dissimilarity=$\frac{∥C∥}{∥P\left(E\right)∥}$.

Since DNA sequences consist of 4 nucleotides {A,T,C,G} = E, then we define a similarity and dissimilarity between the sequences of DNA as:

A similarity =$\frac{∥M_{\mathrm{DNA}}∥}{16}$;

Dissimilarity = $\frac{∥C\left(M_{\mathrm{DNA}}\right)∥}{16}$.

Example 15.

(Continue for Example 9)

(i) 

A similarity =$\frac{∥M_{\mathrm{DNA}}∥}{16}$ = 1; dissimilarity=$\frac{∥C\left(M_{\mathrm{DNA}}\right)∥}{16}=0$.

(ii) 

A similarity =$\frac{∥M_{\mathrm{DNA}}∥}{16}=\frac{12}{16}$; dissimilarity=$\frac{∥C\left(M_{\mathrm{DNA}}\right)∥}{16}=\frac{4}{16}$.

7. Conclusions and Future Work

Matroids are one of the most important branches of modern mathematics, which play an important role in various applications. Determining the existence of mutations of DNA and RNA is an essential issue in biological applications. We created Algorithm 1 to determine the existence (or absence) of a mutation and through Algorithm 1, the gene sequence was converted into a matrix. The mutation of DNA and RNA can be determined by the matrix. The matroidal structures of matrices are used for determining the existence of mutations which is an essential issue in biological applications. In the future, these results can be applied to develop new mutations that are useful in agriculture and industry, as well as the pharmaceutical industry and the treatment of diseases. We will study how matroidal structures can be used to model mutations in RNA secondary structures and predict the effects of these mutations on RNA function, how matroidal optimization can be used to model and predict the evolution and fitness of RNA viruses, how matroidal structures can be used to model DNA recombination and repair processes and predict the effects of mutations on these processes and how matroidal structures can be used to model genome rearrangements and predict the effects of mutations on these processes.