Fully nonlocal stochastic control problems with fractional Brownian motions and Poisson jumps

1.   Introduction

The strings of DNA sequences are shaped from nucleotides which are bonded together. DNA has four nucleotides called guanine $G$, cytosine $C$, adenine $A$ and thymine $T$ (or uracil $U$). $G$ (resp. $A$) is paired with $C$ (resp. $T$ or $U$). The interaction between them is folding. The chain of nucleotides may be folding and bonding, but these interactions only occur under specific energy conditions. Here, we utilize nucleotide chains in conformity with the topological model; see ^[1,10]. A change or metamorphosis is called a mutation. Chromosome and gene alterations, known as mutations in biology, frequently manifest physically. The consequences of a mutation depend on the region where the genetic material's sequence has changed. On the other hand, insertion or deletion mutations might result in the production of gene products that are not functional. Large-scale mutations can also occur, resulting in the inversion, insertion, duplication, deletion, transposition, or translocation of lengthy strands of DNA. A mutation's outcome could be negative, positive, neutral, or even barely noticeable. A mutation may result in the removal or addition of a particular function, altered levels of expression, or even mortality in the developing embryo. A lot of scientists have worked hard to find and fix gene mutations. They have employed a few techniques for investigation, including single-stranded DNA oligonucleotide analysis, single-strand conformation polymorphism analysis, two-dimensional gene scanning, protein truncation testing, and denaturing high performance liquid chromatography ^[5,35]. The use of computer technology for the management of biological data is known as bioinformatics. To collect, store, analyze, and combine biological and family data for use in the discovery and development of gene-based drugs, information processing systems are utilized. The increase of publicly accessible genetic data as a result of the Human Genome Project has sparked the need for bioinformatics capabilities. A virus may cause mutations or the host may edit them, and sequencing mistakes can further complicate matters.

Multiset theory was introduced by Gostelow ^[23]. The concept of a multiset (or bag) is the generalization of a set. A member of a multiset has more than one membership (see ^{[7,8,29,36,37]}); the use of multisets in mathematics predates the name multiset by nearly 90 years.

Topology is a branch of geometry with the name of rubber sheet geometry. It has many real-life applications and solves some problems that are directly or indirectly related to continuity. Its study does not depend on the dimension, i.e., increasing or decreasing can occur without cutting ^[26,31,38]. Using the neighborhood system, graphs have been represented topologically, as in ^[14,32], and some topologies have been represented by neighborhoods and graphs, as in ^[28]. Recently, both graphs and rough sets have been used to represent structures such as self-similar fractals ^[11,15], the human heart ^[12,13,33] and DNA ^[17,18,19], making them useful in physics, medicine and biology ^[2,3,4], respectively.

Graph theory is a mathematical tool to solve some real-life problems. Graphs can be used to model many types of relations and processes in physical, biological ^[30], social and information systems. Many practical problems can be represented by graphs. Many previous studies have investigated the similarity of genetic sequences ^{[20,21,27,34]}.

Our aim with this paper is to examine the existence of gene mutations based on relations, and through the use of metric space, topological structures and graph-based models. We generate a code that depends on the multiset, the relation and the metric space between the sequences of genes (DNA sequences) to determine the presence of the mutation, its locations, if any, and the amino acids. Graph theory is used to determine mutations of genes, and the similarity between DNA sequences will be studied. Finally, we combine the concept of a multiset and association indices to study the similarity between mutations for genes.

2.   Preliminaries

In what follows, a short survey of multisets and the corresponding theories will be given based on the work of Yager in ^[39]. Also, we introduce a short survey on genetic mutations, as described in ^[5,37], and some methods for mutation analysis and mutation detection are mentioned.

Definition 2.1. ^[9,24,39] A multiset $M$ assigned from a nonempty set $X$ and presented by a function $C_{M}(x): X \rightarrow N$, where $N$ denotes the natural numbers. $C_{M}(x)$ represents the number of element $x$ which occurs in $M$. In other words, $M$ from $X = \{x_{1}, x_{2},$ $\cdots, x_{n}\}$ to $N$ is written as $M =$ $\{\frac{m_{1}}{x_{1}}, \frac{m_{2}}{x_{2}},$ $\cdots,$ $\frac{m_{n}}{x_{n}}\}$, where $m_{i}$ is the number of $x_{i}$, for $i$ $= 1, 2, \cdots, n$ that can occur in $M$.

Proposition 2.2. Let $M$ and $N$ be two multisets assigned on $X$. Then, the following holds

(i) $M = N$ if $C_{M}(x) = C_{N}(x)$ $\forall$ $x \in X$.

(ii) $M \subseteq N$ if $C_{M}(x)\leq C_{N}(x)$ $\forall$ $x \in X$.

(iii) $W =$ $M \cup N$ if $C_{W}(x) =$ $Max \{C_{M}(x), C_{N}(x)\}$ $\forall$ $x \in X$.

(v) $W =$ $M \ominus N$ if $C_{W} (x) =$ $Max\{C_{M}(x)- C_{N}(x), 0\}$ $\forall$ $x \in X$.

(vi) $W =$ $M \oplus N$ if $C_{W}(x) =$ $C_{M}(x) + C_{N}(x)$ $\forall$ $x \in X$.

(vii) $W =$ $M \cap N$ if $C_{W}(x) =$ $Min\{C_{M}(x), C_{N}(x)\}$ $\forall$ $x \in X$.

Here, $\ominus$ and $\oplus$ denote a multiset subtraction and a multiset addition, respectively. It is noted that any set is a special case of multiset.

Definition 2.3. ^[22] Let $K_{1}$ and $K_{2}$ be two multisets assigned on $X$, and have $C_{K_{1}}$ and $C_{K_{2}}$, respectively. The Cartesian product of $K_{1}$ and $K_{2}$ is defined by $K_{1}\times K_{2} =$ $\{\frac{(\frac{m}{x}, \frac{n}{y})}{mn}:$ $x\in^{m} K_{1}, y \in^{^{n}} K_{2}\}$.

Definition 2.4. ^[22] A submultiset $\mathcal{R}$ of $M \times M$ is said to be a multiset relation on $M$, if for each $(\frac{m}{x}, \frac{n}{y})$ of $\mathcal{R}$, there is a product of $C_{1}(x, y)$ and $C_{2}(x, y)$ which can be counted. The relationship between $\frac{m}{x}$ and $\frac{n}{y}$ can be formulated as $\frac{m}{x} \mathcal{R} \frac{n}{y}$.

In graph theory ^[10], the set of vertices will be denoted by $V$ of a finite set. The set of edges have the form $E(V) =$ $\{ \{u, v\}$ s. t. $u, v\in V$ $u\neq v\}$. In other words, $u, v$ are called adjacent vertices. In this paper, the graph will be denoted by $G = (V, E)$, $V_{G}$ is the vertex of $G$ and $E_{G}$ is the set of edges. The graph $G = (V_{G}, E_{G})$ is a directed graph if each edge has a direction. The Łkaszyk-Karmowski distance function ^[43] is a function defining a distance between two random variables or two random vectors. The axioms of this function are as follows:

● $d(x, y) > 0$,

● $d(x, y) = d(y, x)$,

● $d(x, z)\leq d(x, y)+d(y, z)$.

3.   Mutations from the viewpoint of multisets, relations and metric spaces

In this section, the concepts of multisets, metric spaces and multiset relations are applied in MSC code building. The MSC code determines the existence of a mutation locations and number of mutations; it also identifies amino acids. The MSC code specifies the number of (A, T, G, C) elements, the relation between them, the places of their difference their numbers, and the different amino acids. Also, the code can be used to study the similarity between the DNA sequences (numbers of elements, matches and mismatches).

Definition 3.1. Let $M^{3^{'}}_{5^{'}}$ be a sense strand of DNA and $M^{5^{'}}_{3^{'}}$ be an antisense strand of DNA. Define a multiset of DNA sequence as $M = \{\frac{m_{i}}{x}: x\in \{A, T, G, C\}$, where $m_{i}$ is the time of occurring for $x \}$

Remark 3.2. In Definition 3.1, there are two DNA multisets $M_{1}$ and $M_{2}$ for $M^{3^{'}}_{5^{'}}$ and $M^{5^{'}}_{3^{'}}$, respectively.

Definition 3.3. Let $M_{1}$ and $M_{2}$ be DNA multisets. Define the DNA Cartesian product of $M_{1}$ and $M_{2}$ by $M_{1}\times M_{2} =$ $\{\frac{(\frac{m}{x}, \frac{n}{y})}{mn}:$ $x\in^{m}M_{1}, y \in^{^{n}}M_{2}\}$.

Definition 3.4. Let $M_{1}$ and $M_{2}$ be DNA multisets. Define multibinary relation $R\subseteq M_{1}\times M_{2} =$ $\{{(\frac{m}{x}, \frac{n}{y})}:$ $x\in^{m}M_{1}, y \in^{^{n}}M_{2}\}$.

Definition 3.5. Let $M_{1}$ and $M_{2}$ be DNA multisets. Define the correlation coefficient between $M_{1}$ and $M_{2}$ as $C_{M_{1}, M_{2}} = \frac{1}{\left|M_{1}\right|\left|M_{2}\right|}\sum\limits_{x_i\in M_{1}, y_i\in M_{2}}{C(x_i)C(y_i)}$, where $C(x_i)$ and $C(y_i)$ are time of occurring for $x_{i}$ and $y_{i}$ in $M_{1}$ and $M_{2}$, respectively.

Corollary 3.6. $C_{M_{1}, M_{1}} = \frac{1}{{|M_{1}|}^{2}}\sum\limits_{x_i\in M_{1}}{(C(x_i))}^{2}.$

Corollary 3.7. $0 < C_{M_{1}, M_{2}}\le 1$.

Proof. Since $M_{1}\ne \phi \to |M_{1}|\ne 0$, $M_{2}\ne \phi \to |M_{2}|\ne 0$ and $|M_{1}| \ge C_{M_{1}}(x_i)$, $|M_{2}|\ge C_{M_{1}}(y_i)$, then $|M_{1}||M_{2}|\ge \sum{C_{M_{1}}(x_i)C_{M_{2}}(y_i)}$. Therefore, $0 < C_{M_{1}, M_{2}}\le 1$. □

Remark 3.8. $n(M_{1})$ is the number of elements existing in DNA multiset $M_{1}$, and $n(M_{1}) = 4$ at most.

Definition 3.9. Let $M_{1}$ and $M_{2}$ be DNA multisets, $n(M_{1}) =$ $n(M_{1})$. Define the distance function between $M_{1}$ and $M_{2}$ as $d_{DNA}(M_{1}, M_{2}) =$ $\frac{C_{M_{1}, M_{2}}} {\sqrt{{C_{M_{1}, M_{1}}} \times C_{M_{2}, M_{2}}}}.$

Remark 3.10. From Definition 3.9, $d_{DNA}(M_{1}, M_{2}) =$ $\frac{\sum\limits_{x_i, y_i} C(x_i) C(y_i)} {\sqrt{\sum_{x_i}(C_{M_{1}} (x_i))^{2} \times \sum_{x_i}(C_{M_{2}}(y_i))^{2}}}.$

Theorem 3.11. $d_{DNA}(M_{1}, M_{2})$ is associated with the following axioms:

(i) $d_{DNA} > 0$,

(ii) $d_{DNA}(M_{1}, M_{1}) = 1$,

(ii1) $d_{DNA}(M_{1}, M_{2}) = d_{DNA} (M_{2}, M_{1})$,

(iv) $d_{DNA}{(M_{1}, M_{3})}+d_{DNA}{(M_{3}, M_{2})}\geq d_{DNA}{(M_{1}, M_{2})}$.

Proof. (i) By Corollaries 3.6 and 3.7, $0 < C_{M_{1}, M_{2}}\le 1$, $C_{M_{1}, M_{1}} > 0$ and $C_{M_{2}, M_{2}} > 0$. Then, $d_{DNA} > 0$.

(ii) By Corollary 3.6, $d_{DNA}(M_{1}, M_{1}) = 1$.

(iii) By Definition 3.5, $d_{DNA}(M_{1}, M_{2}) =$ $\frac{C_{M_{1}, M_{2}}}{\sqrt{C_{M_{1}, M_{1}}\times {C_{M_{2}, M_{2}}}}}$ = $\frac{C_{M_{2}, M_{1}}}{\sqrt{C_{M_{2}, M_{2}}\times {C_{M_{1}, M_{1}}}}} =$ $d_{DNA} (M_{2}, M_{1})$.

(iv) Since $|M_{1}|$ $\ge$ $C_{M_{1}}(x_i)$, $|M_{2}|$ $\ge$ $C_{M_{2}}(y_i)$ and $|M_{3}|$ $\ge$ $C_{M_{3}}(z_i)$, then, by Definition 3.5, $C_{M_{1}, M_{2}} =$ $\frac{1}{|M_{1}||M_{2}|}$ $\sum \limits_{x_i\in M_{1}, y_i\in M_{2}}{C(x_i)C(y_i)},$ $C_{M_{2}, M_{3}} =$ $\frac{1}{|M_{2}||M_{3}|}$ $\sum\limits_{z_i\in M_{3}, y_i\in M_{2}}{C(z_i)C(y_i)}$ and $C_{M_{1}, M_{3}} =$ $\frac{1}{|M_{1}||M_{3}|} \sum\limits_{x_i\in M_{1}, z_i\in M_{3}}{C(x_i)C(z_i)}$. To prove that $d_{DNA}{(M_{1}, M_{3})}+d_{DNA}{(M_{3}, M_{2})}\geq d_{DNA}{(M_{1}, M_{2})}$, it is sufficient to prove that $\frac{C_{M_{1}, M_{2}}} {\sqrt{C_{M_{1}, M_{1}} \times {C_{M_{2}, M_{2}}}}}$ $\leq$ $\frac{C_{M_{1}, M_{3}}}{\sqrt{C_{M_{1}, M_{1}}\times {C_{M_{3}, M_{3}}}}}+$ $\frac{C_{M_{3}, M_{2}}}{\sqrt{C_{M_{3}, M_{3}}\times {C_{M_{2}, M_{2}}}}}$. Since $C_{M_{1}, M_{3}}+$ $C_{M_{3}, M_{2}} =$ $\frac{1}{|M_{1}||M_{3}|}$ $\sum\limits_{x_i\in M_{1}, z_i\in M_{3}} {C(x_i)C(z_i)}+$ $\frac{1}{|M_{3}||M_{2}|}$ $\sum\limits_{z_i\in M_{3}, y_i\in M_{2}}{C(z_i)C(y_i)} =$ $\frac{1}{|M_{2}||M_{1}||M_{3}|}$ $|M_{2}|$ $\sum\limits_{x_i\in M_{1}, z_i\in M_{3}}$ ${C(x_i)C(z_i)}+$ $\frac{1}{|M_{1}||M_{3}||M_{2}|}$ $|M_{1}|$ $\sum\limits_{z_i\in M_{3}, y_i\in M_{2}}$ ${C(z_i)C(y_i)}$ $\ge$ $\frac{1}{|M_{2}||M_{1}||M_{3}|}$ $|M_{2}|$ $\sum\limits_{x_i\in M_{1}, z_i\in M_{3}}{C(x_i)C(z_i)}$ $\ge$ $\frac{1}{|M_{2}||M_{1}||M_{3}|}$ $\sum\limits_{y_i\in M_{2}}{C(y_i)}$ $\sum\limits_{x_i\in M_{1}, z_i\in M_{3}}{C(x_i)C(z_i)}$ $\ge$ $\frac{1}{|M_{1}||M_{2}|}$ $\sum\limits_{x_i\in M_{1}, y_{i}\in M_{2}}{C(x_i)C(y_i)} =$ $C_{M_{1}, M_{2}}$, where $|M_{2}||M_{1}|$ $\leq$ $\sum{(C(x_{i}))}^{2}$ $\times$ $\sum{(C(y_{i}))}^{2},$ $|M_{2}||M_{3}|$ $\leq$ $\sum{(C(y_{i}))}^{2}$ $\times$ $\sum{(C(z_{i}))}^{2},$ $|M_{3}||M_{1}|$ $\leq$ $\sum{(C(z_{i}))}^{2}$ $\times \sum{(C(x_{i}))}^{2}$, $|M_{2}||M_{1}|$ $\geq$ ${\sqrt{\sum{(C(x_{i}))}^{2}\times \sum{(C(y_{i}))}^{2}}},$ $|M_{2}||M_{3}|$ $\geq$ ${\sqrt{\sum{(C(y_{i}))}^{2} \times \sum{(C(z_{i}))}^{2}}}$ and $|M_{3}||M_{1}|$ $\geq$ ${\sqrt {\sum{(C(z_{i}))}^{2}\times \sum{(C(x_{i}))}^{2}}}$. Therefore, $|M_{2}||M_{1}|$ $\geq{\sqrt{|M_{2}||M_{1}|}}$ $\geq{\sqrt{C_{M_{1}, M_{1}} \times {C_{M_{2}, M_{2}}}}}$. □

Remark 3.12. We can call the function $d_{DNA}$ a DNA metric space.

Remark 3.13. Theorem 3.11 satisfies the condition of Łkaszyk-Karmowski distance.

Proposition 3.14. The distance function $1-d_{DNA}$ is a metric space.

Proof. Refer to Theorem 3.11. □

Theorem 3.15. If $d_{DNA}{(M_{1}, M_{2})}$ = 1, then there is no mutation.

Proof. Let $M_{1} =$ $\{n_{1}/G, n_{2}/A, n_{3}/T, n_{4}/C\}$, $M_{2} =$ $\{m_{1}/C, m_{2}/T, m_{3}/A, m_{4}/G\}$ and $d_{DNA}{(M_{1}, M_{2})}$ = 1. Then, by Theorem 3.11, we get that $d_{DNA} (M_{1}, M_{2}) =$ $1$ $= \frac{C_{M_{1}, M_{2}}} {\sqrt{C_{M_{1}, M_{1}}} \times {C_{M_{2}, M_{2}}}}$. Then, $C_{M_{1}, M_{2}} =$ $\sqrt{C_{M_{1}, M_{1}}}$ implies that $C_{M_{1}, M_{2}}^{2} =$ $C_{M_{1}, M_{1}}$ $\times$ $C_{M_{2}, M_{2}}$. Using Definition 3.5, $(n_{1}m_{1}+n_{2}m_{2}+n_{3}m_{3}+n_{4}m_{4})^{2}$ = $(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2})$ $\cdot$ $(m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2})$. Then, $n_1 = \lambda m_1$, $n_2 = \lambda m_2$, $n_3 = \lambda m_3$ and $n_4 = \lambda m_4$. So, if $\lambda = 1$, then $n_{1} = m_{1}$, $n_{2} = m_{2}$, $n_{3} = m_{3}$ and $n_{4} = m_{4}$. This means that there is no mutation. □

Corollary 3.16. From Theorem 3.15, we have that $d_{DNA}{(M_{1}, M_{2})} \neq1$; then, there is a mutation.

We present the MSC code in the Appendix as an algorithm which is used to generate multisets, relations and a metric space between $M_{1}$ and $M_{2}$. Some examples are given to illustrate the proposed results and MSC code algorithm.

Example 3.17. Arabidopsis thaliana gamma-glutamylcysteine synthetase gene (abbr. CAD2) ^[44]

$5^{'}\; \; ATCGATATGTAACACAAT \cdots TGTATGTTTTT\; \; 3^{'}$;

$3^{'}\; \; TAGCTATACATTGTGTTA \cdots ACATACAAAAA\; \; 5^{'}$. Using the MSC code algorithm, we have $M_{1} =$ $\{\frac{1019}{G}, \frac{1543}{A}, \frac{1859}{T}, \frac{856}{C}\}$, $\left|M_{1}\right| =$ $5277$;

$M_{2} =$ $\{\frac{1019}{C}, \frac{1543}{T}, \frac{1859}{A}, \frac{856}{G}\}$, $\left|M_{2}\right| =$ $5277$.

The distance between $M_1$ and $M_2$ equals $1$ (no mutation). The relation between $M_1$ and $M_2$ according to their MSC code, is described in Table 1. The relation between $M_1$ and $M_2$ is

$\mathcal{R} =$ $\{(\frac{1859}{T}, \frac{1859}{A}), (\frac{1543}{A}, \frac{1543}{T}), (\frac{1019}{G}, \frac{1019}{C}), (\frac{856}{C}, \frac{856}{G})\}$. This relation indicates no mutation.

Example 3.18. If we do a mutation in CAD2 ^[44] in Example 3.1. Using the MSC code algorithm, we have $M_1 =$ $\{\frac{1014}{G}, \frac{1539}{A}, \frac{1859}{T}, \frac{860}{C}\}$, $\left|M_1\right| =$ $5272$; $M_2 =$ $\{\frac{1014}{C}, \frac{1543}{T}, \frac{1859}{A}, \frac{856}{G}\}$, $\left|M_2\right| =$ $5272$.

The distance between $M_1$ and $M_2$ equals $0.9999979580282978$ according to the MSC code; the result was a mutation and this corresponds to data reported by the National Center for Biotechnology Information (NCBI) ^[44]. The position of the mutation is

$[2568, 2578, 2595, 2609, 2639, 5076]$;

$[C, \; \; \; \; T, \; \; \; ,\; \; \; \; \; C, \; \; \; \; C, \; \; ,\; \; \; \; \; \; G, \; \; \; \; C]$;

$[T, \; \; \; \; T, \; \; \; \; C, \; \; \; \; T, \; \; \; \; A, \; \; \; \; T]$.

The amino acid resulting from the mutation is presented in Table 2. The relation between $M_1$ and $M_2$ is outlined in Table 2. The relation between $M_{1}$ and $M_{2}$ is $\mathcal{R} =$ $\{(\frac{1}{G},$ $\frac{1}{A}),$ $(\frac{1}{T},$ $\frac{1}{T}),$ $(\frac{3}{C},$ $\frac{3}{T}),$ $(\frac{1}{C},$ $\frac{1}{C}),$ $(\frac{1539}{A},$ $\frac{1539}{T}),$ $(\frac{1858}{T},$ $\frac{1858}{A}),$ $(\frac{1013}{G},$ $\frac{1013}{C}),$ $(\frac{856}{C},$ $\frac{856}{G})\}$ according to their MSC code as described in Table 3. This relation indicates a mutation.

Corollary 3.19. From Theorem 3.15, the relation will be $R =$ $\{(n_{1}/G, n_{1}/C)/n_{1}n_{1},$ $(n_{2}/A, n_{2}/T)/n_{2}n_{2},$ $(n_{3}/T, n_{3}/A)/n_{3}n_{3},$ $(n_{4}/C, n_{4}/G)/n_{4}n_{4}\}$.

Proposition 3.20. Let $R$ be a relation between DNA multisets $M_{1}$ and $M_{2}$. Then, if $R$ is either reflexive or transitive, then there is a mutation.

Proof. Suppose that $M_{1} =$ $\{\frac{n_1}{x}: x\in \{A, T, G, C\}$, where $n_1$ is the number occurrences of $\{x\} =$ $\{n_1/G, n_2/A, n_3/T, n_4/C\}$, $M_{2} =$ $\frac{m_1}{y}:$ $y\in \{A, T, G, C\}$, where $m_1$ is the number occurrences of $\{y\} = \{m_1/C, m_2/T, m_3/A, m_{4}/G\}$ and $R = \{(\frac{m}{x}, \frac{n}{y})\}: x \in^{m} M_{1}, y \in^{n} M_{2}\}$. Then, every $C$ is linked with $G$. On the other side, $A$ is linked with $T\equiv U$. Otherwise, a mutation will be occurred. □

4.   Topological structures of DNA mutations

In the study of the congruence and determination of the presence of mutations between DNA sequences, we have four bases $\{A, T, G, C\}$; thus, we have the $12$ mutation rates $A \rightarrow C$, $A \rightarrow G$, $\cdots$, $T \rightarrow G$ at a particular site. In the study of the similarity between the DNA sequences, we have $12$ difference rates $A \rightarrow T$, $A \rightarrow G$, $\cdots$, $T \rightarrow A$ at a particular site. We can use these rates to study SARS-CoV-2 through the mutation of its genes. So, our study can yield a model of the pattern of mutations in SARS-CoV-2 and the alternative model for the mutations that occur in SARS- CoV-2 can be developed. If the length $n_{i, j} > 0$, then we suggest $S_{i} = \{A, T, C, G\}$, which has more than an average likelihood of linking to $S_{j}$. For each $i = 1, \cdots, 4$, define $\mathcal{R}_{i} =$ ${S_{i}} \cup {S_{j}}$, $n_{i, j} > 0$. The collection $\mathcal{R}_{0} =$ $\{R_{i}\}^{4}_{i}$ is not itself a topology, but we extend it to one, defining $0$ to be a minimal topological structure on genotypes containing $\mathcal{R}_{0}$. The topological space $\tau_{0}$ will be generated by a basis induced by a finite intersection of the sets in $\mathcal{R}_{0}$. The topological structure is referred as a mutation space and is called a mutation topological structure.

In this section, we use the proposed MSC code algorithm, the following definition is given.

Definition 4.1. Let $X$ be the set of nucleotides of a DNA sequence such that $X = \{A, T, G, C\}$, and let there exist a bonding between $x_{i}, MSC y_{j}$ in $X$, referred to as $n(x_{i}, y_{j})\neq 0$. Otherwise, $n(x_{i}, y_{j}) = 0$.

Definition 4.2. Let $X$ be the set of nucleotides of a DNA sequence. Define a relation $R^* = \{(x, y):$ $n(x, y)\neq 0, x, y\in X\}$.

We state some properties for the cases of mutations and no mutation.

(i) If there is no mutations, then

● $R^*$ is not reflexive,

● $R^*$ is symmetric,

● $R^*$ is transitive.

(ii) If there is a mutation, then $R^*$ may be reflexive, symmetric and transitive.

Example 4.3. {NM 000518.4 Homo sapiens hemoglobin subunit beta (abbr. HBB), mRNA ^[44] sequence: HBB gene range: $1$ to $626$};

$5^{'}\; \; ACATTTGCTT \cdots CATTGC\; \; 3^{'}$;

$3^{'}\; \; TGTAAACGAA \cdots GTAACG\; \; 5^{'}$;

$M_{1} =$ $\{\frac{157}{G},$ $\frac{167}{A},$ $\frac{137}{T},$ $\frac{165}{C}\},$ $\left|M_{1}\right| =$ $626$;

$M_{2} =$ $\{\frac{157}{C},$ $\frac{167}{T},$ $\frac{137}{A},$ $\frac{165}{G}\},$ $\left|M_{2}\right| =$ $626$.

The relation between $M_{1}$ and $M_{2}$ according to the MSC code is described in Tables 4 and 5. $\mathcal{R}^* =$ $\{(T, A),$ $(A, T),$ $(G, C),$ $(C, G)\}$. This relation indicates no mutation and is consistent with the report by the NCBI ^[44].

Example 4.4. If we do a mutation in CAD2, then $M_{1} =$ $\{\frac{1012}{G},$ $\frac{1541}{A},$ $\frac{1856}{T},$ $\frac{856}{C}\},$ $\left|M_{1}\right| =$ $5265$, and $M_{2} =$ $\{\frac{1012}{C},$ $\frac{1540}{T},$ $\frac{1858}{A},$ $\frac{855}{G}\},$ $\left|M_{2}\right| =$ $5265$. The position of the mutation $[17,686, 5073]$ is $[G, \; \; C, \; \; A]$ and $[A, \; \; A, \; \; C]$. The amino acid which results from the mutation is presented in Table 6. The distance between $M_{1}$ and $M_{2}$ equals $0.999999362175112$, according to the MSC code. The relation between $M_{1}$ and $M_{2}$ according to the MSC code, is described in Tables 7 and 8. $\mathcal{R}^* =$ $\{(T, A),$ $(C, A),$ $(G, A),$ $(A, T),$ $(A, C),$ $(G, C),$ $(C, G)\}$. So, there is a mutation.

Next, a topological structure will be defined in terms of $R^{*}$.

Definition 4.5. Let $R^{*}$ be a relation on $X$. Define a subbase $S =$ $\{x R^{*}: x\in X \}$ for some topology $\tau_{DNA}$ on $X$.

Example 4.6. (continued from Example 4.3) $S =$ $\{A R^{*},$ $T R^{*},$ $C R^{*},$ $G R^{*}\} =$ $S =$ $\{\{A\},$ $\{T\}, \{C\},$ $\{G\}\}$. Then, the base $\beta$ will be $\{X, \{A\},$ $\{T\}, \{C\},$ $\{G\}\}$ and $\tau_{DNA} =$ $\{X, \phi, \{A\},$ $\{T\},$ $\{C\},$ $\{G\},$ $\{A, T\},$ $\{T, C\},$ $\{C, G\},$ $\{G, A\},$ $\{A, C\},$ $\{T, G\},$ $\{A, T, C\},$ $\{A, T, G\},$ $\{T, C, G\},$ $\{A, G, C\}\}$. Therefore, this space is discrete. This means that, if every subset of $X =$ $\{A, T, C, G\}$ is open and closed, then there is no mutation.

Example 4.7. (continued from Example 4.4) $S =$ $\{\{T, C\},$ $\{A\},$ $\{A, G\},$ $\{A, C\}\}$. Therefore, a base $\beta =$ $\{X,$ $\{T, C\},$ $\{A\},$ $\{A, G\},$ $\{A, C\},$ $\{C\}\}$ and $\tau_{DNA} =$ $\{X,$ $\phi,$ $\{T, C\},$ $\{A\},$ $\{A, G\},$ $\{A, C\},$ $\{C\},$ $\{A, T, C\},$ $\{A, C, G\}\} \equiv$ general topology.

Now, the existence of a mutation will be determined based on the type of topological structure.

Proposition 4.8. If the DNA sequence has a mutation, then the generated topology is a general topological structure.

Proof. Let $\tau_{DNA}$ be a class of sets on $X$ generated by the mutation of DNA sequences. Consider that $\tau_{DNA} =$ $\{G: G =$ $\bigcup_{i} (\bigcap^{n}_{j}A_{_{ij}}),$ $A_{ij} \in S\}$ is a class of sets of $X$. Now, it is sufficient to prove that $\tau_{DNA}$ is a topological structure.

(i) $\cap^{n}_{j\in \phi}A_{_{ij}} = X\in \tau_{DNA}$ and $\bigcup_{i\in \phi}(\bigcap^{n}_{j}A_{_{ij}}) = \phi \in \tau_{DNA}$.

(ii) $G_1, G_2, \cdots G_n \in \tau_{DNA}$; then, $G_1 = \bigcup_{i_{1}}(\bigcap^{n}_{j_{1}}A_{_{i_{1}j_{1}}})$, $G_2 = \bigcup_{i_{2}}(\bigcap^{n}_{j_{2}}A_{_{i_{2}j_{2}}}), \cdots, G_n = \bigcup_{i_{n}}(\bigcap^{n}_{j_{n}}A_{_{i_{n}j_{n}}})$. $G_1 \cap G_2 \cap \cdots \cap G_n =$ $\bigcup_{i_{1}, i_{2} \cdots i_{n}} (\bigcap\limits^{n}_{j_{1}} A_{_{i_{1} j_{1}}}) \cap \bigcap\limits^{n}_{j_{2}} A_{_{i_{2} j_{2}}} \cap \cdots \cap \bigcup_{{i}_{n}}\bigcap^{n}_{j_{n}}A_{_{i_{n}j_{n}}}) =$ $\bigcup_{i_{1}, i_{2}, \cdot, i_n} (\bigcap\limits^{n}_{k_{1}} B_{k_{1} j_{k}})$, where $B_{kj_k} = A_{i_{1}j_{1}}\cap A_{i_{2}j_{2}}\cap \cdots \cap A_{i_{n}j_{n}}$. Since each $A_{i_{1}j_{1}}, A_{i_{2}j_{2}} \cdots A_{i_{n}j_{n}}\in S$, $B_{kj_k}\in \beta$; therefore, $G_1 \cap G_2\cap \cdots \cap G_n \in \tau_{DNA}$.

(ii) $G_1, G_2, \cdots G_n \cdots \in \tau_{DNA}$; then, $G_1 \cup G_2 \cup \cdots \cup G_n \cup \cdots =$ $\bigcup\limits_{i_{1}, i_{2} \cdots i_{n}}(\bigcap\limits^{n}_{j_{1}} A_{i_{1} j_{1}}$ $\cup$ $\bigcap\limits^{n}_{j_{2}} A_{i_{2} j_{2}}$ $\cup \cdots \cup$ $\bigcup_{i_n} \bigcap\limits^{n}_{j_{n}} A_{_{i_{n}j_{n}}}\cup \cdots)$. Hence, $G_1 \cup G_2 \cup \cdots \cup G_n \cup \cdots \in \tau_{DNA}$. □

Proposition 4.9. If there is no mutation in DNA, then $\tau_{DNA}$ is a discrete topology.

Proof. Suppose that the DNA sequence has no mutation. Then, $n(x, x) = 0$ and $n(x, y)\neq 0$ $\forall$ $x, y \in X$. Hence, $x R^{*} =$ $\{\{y\}:$ $\forall$ $y\in X\}$. This means that $A \rightarrow T$, $C \rightarrow G$, $T \rightarrow A$, $G \rightarrow C$. So, $S =$ $\{\{y\} :y\in X\}$. Therefore, $\tau_{DNA}$ is discrete. □

The converse of Proposition 4.9 may not be true, in general.

Example 4.10. $S = \{\{T\}, \{A\},$ $\{C, G\}, \{C\}\}$ is a subbase for a discrete topological structure. But, there is a mutation because $G$ is bonded with $C$.

Example 4.11. Let a DNA sequence be $5^{'}\; \; ACGT\; \; 3^{'}$ and $3^{'}\; \; GATC\; \; 5^{'}$. Then, $S =$ $\{\{G\},$ $\{A\},$ $\{T\},$ $\{C\}\}$ and $\tau_{DNA}$ is a discrete topology. But, there is a mutation, as no gene consists of only four nucleotides (length 4 only or a little more); the length of a gene is measured in kilobytes.

Corollary 4.12. If the topological structure generated from the DNA sequences has

(1)- a general topological structure, then there is a mutation;

(2)- a discrete topology, then there may or may not be a mutation. We use Theorem 3.15, Corollary 3.16 and Proposition 3.20 to figure out if there is a mutation.

5.   Mutations by similarity of genes based on the association indices and multisets

Bass et al. ^[6] provided an overview of commonly used association indices, including the Jaccard index and the Pearson correlation coefficient, and compared their performance on different types of analysis for a biological network. An association index is a measure that quantifies interaction profile similarity. They discussed the differences and similarities between association indices. There exist many association indices:

(i) The Jaccard index: $J_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{|N_{A}\cup N_{B}|}$;

(ii) The Simpson index: $S_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{Min \{|N_{A}|, |N_{B}|\}}$;

(iii) The geometric index: $G_{AB} =$ $\frac{|N_{A}\cap N_{B}|^{2}} {|N_{A}|. |N_{B}|}$;

(iv) The cosine index: $C_{AB} = \frac{|N_{A}\cap N_{B}|}{\sqrt{|N_{A}\cdot N_{B}|}}$.

We apply association indices to determine whether there is a mutation by calculating the association indices for each pair of nucleotides $\{A, T, G, C\}$ of the gene. If the associations are zero, there is no mutation; otherwise, there is a mutation.

Example 5.1. (continued from Example 4.3)

$e_{1} =$ $167 =$ $e_{5}$, $e_{2} =$ $157 =$ $e_{7}$, $e_{3} =$ $165 =$ $e_{6}$, $e_{4} =$ $137 =$ $e_{8}$.

Let X-type $= \{A\}$, Y-type $= \{T\}$.

Then, we have the following:

Jaccard = $\frac{0}{2} = 0$, Simpson = $\frac{0}{1} = 0$, Geometric = $\frac{0}{1} = 0$, Cosine = $\frac{0}{1} = 0$. Then, Jaccard = Simpson = Geometric = Cosine = $0$. This is for all nucleotides $\{A, T, G, C\}$. Hence, there is no mutation. This is shown in Figures 1–3.

Example 5.2. (continued from Example 4.4)

Let X-type = $\{A\}$, Y-type = $\{G\}$.

Since $e_{1} =$ $1540 =$ $e_{5}$, $e_{2} = 1011 =$ $e_{7}$, $e_{3} = 855 = e_{6}$ and $e_{4} = 1856 = e_{8}$, $e_{9} = 1 = e_{12}$, $e_{10} = 1 = e_{13}$ and $e_{11} = 1 = e_{14}$, we have the following:

Jaccard = $\frac{1}{2},$

Simpson = $\frac{1}{1}$,

Geometric = $\frac{1}{2},$

Cosine = $\frac{1}{\sqrt{2}}.$

Then, Jaccard, Simpson, Geometric and Cosine $\neq 0$. This is for all nucleotides ${A, T, G, C}$. Hence, there is a mutation. This is shown in Figures 4–7.

Note that the degree of a node $A$, say, $|N_A|$, is defined as the number of nodes with which it interacts and $|N_A \cap N_B|$ is the shared partners. Biological processes are implemented through complex interaction networks. Metrics known as association indices can be used to quantify the similarity between genes through the use of a multiset. So, $|N_A|$ is the cardinality of a multiset $M$. Then, the similarity association indices become

$M J_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{|N_{A}\cup N_{B}|}$, $M S_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{Min \{|N_{A}|, |N_{B}|\}}$, $M G_{AB} =$ $\frac{|N_{A}\cap N_{B}|^{2}}{|N_{A}|\cdot|N_{B}}$ and $M C_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{\sqrt{|N_A|\cdot |N_B|}}$.

The dissimilarity association indices are as follows

$M* J_{AB} = 1-\frac{|N_{A}\cap N_{B}|}{|N_{A}\cup N_{B}|}$, $M* S_{AB} = 1-\frac{|N_{A}\cap N_{B}|}{Min \{|N_{A}|, |N_{B}|\}}$, $M* G_{AB} = 1-\frac{|N_{A}\cap N_{B}|^{2}}{|N_{A}|\cdot|N_{B}}$ and $M* G_{AB} = 1-\frac{|N_{A}\cap N_{B}|}{\sqrt{|N_A|\cdot |N_B|}}$.

Remark 5.3. (i) $M_{AB}\geq M_{AB}$.

(ii) If $|N_A| = |N_B|$, then $M_{AB} = M_{AB}$.

Theorem 5.4. The similarity association indices are DNA metric spaces.

Theorem 5.5. The dissimilarity association indices are metric spaces.

Example 5.6. (continued from Example 5.1)

Let X-type $= \{A\}$, Y-type $= \{T\}$; also, $e_{1}$, $e_{8}$ and $e_{8} = 137$.

$N_{A} = \{\frac {e_{1}}{T}, \}$, $N_{B} = \{\frac {e_{8}}{A}\}$.

Jaccard = $\frac {0}{304} = 0$,

Simpson = $\frac{0} {137} = 0$,

Geometric = $\frac {0} {22879} = 0$,

Cosine = $\frac{0} {151.26} = 0$,

Then, Jaccard = Simpson = Geometric = Cosine = $0$. This is for all nucleotides $\{A, T, G, C\}$ since $e_{1} = 1540 = e_{5}$, $e_{2} = 1011 = e_{7}$, $e_{3} = 855 = e_{6}$ and $e_{4} = 1856 = e_{8}$, $e_{9} = 1 = e_{12}$, $e_{10} = 1 = e_{13}$; $e_{11} = 1 = e_{14}$. Hence, there is no mutation. This is shown in Figure 8.

Example 5.7. (continued from Example 5.2)

Let X-type = $\{A\}$, Y-type = $\{G\}$, $e_{1} =$ $1540$, $e_{6} =$ $855$ and $e_{9}$ = $1$. $N_{A} =$ $\{\frac {e_{1}}{T}, \frac {e_{9}}{C}\}$, $N_{B} = \{\frac {e_{6}}{C}\}$.

$M J_{AB} = \frac{|N_{A}\cap N_{B}|}{|N_{A}\cup N_{B}|}$ = $\frac {|\{\frac {e_{1}}{T}, \frac {e_{9}}{C}\}\cap \{\frac {e_{6}}{C}\}|}{|N_{A}\cup N_{B}|}$ = $\frac{|min\{e_{9}, e_{6}\}|}{|max\{\{e_{6}, e_{9}\}, e_{1}\}}$ = $\frac{|\{e_{9}\}|}{|\{e_{6}, e_{1}\}}$ = $\frac {1} {2395} \neq 0$,

$M S_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{Min \{|N_{A}|, |N_{B}|\}}\neq 0$,

$M G_{AB} =$ $\frac{|N_{A}\cap N_{B}|^{2}}{|N_{A}|\cdot|N_{B}}\neq 0$,

$M C_{AB} =$ $\frac{|N_{A}\cap N_{B}|}{\sqrt{|N_A|\cdot |N_B|}}\neq 0$. Additionally, $e_{1} = 1540 = e_{5}$, $e_{2} = 1011 = e_{7}$, $e_{3} = 855 = e_{6}$; $e_{4} = 1856 = e_{8}$, $e_{9} = 1 = e_{12}$, $e_{10} = 1 = e_{13}$ and $e_{11} =$ $1 =$ $e_{14}$.

Then, Jaccard = Simpson = Geometric = Cosine $\neq 0$. This is for all nucleotides $\{A, T, G, C\}$. Hence, there is a mutation. This is shown in Figure 9.

Example 5.8. (A similarity and dissimilarity between the sequences of DNA)

Let $GATACCCCCCGG,$ $GATACGACCCGG,$ $GATACGCCCCGG,$ $CATACGACTCGG$ and $GATAGACTCGG$ be five sequences for DNA. Then, $A =$ $\{\frac{3}{G},$ $\frac{2}{A},$ $\frac{1}{T},$ $\frac{6}{C}\}$, $\left|A\right| = 12$, $B =$ $\{\frac{4}{C},$ $\frac{1}{T},$ $\frac{3}{A},$ $\frac{4}{G}\}$, $\left|B\right| =$ $12$, $C =$ $\{\frac{5}{C},$ $\frac{1}{T}, \frac{2}{A},$ $\frac{4}{G}\},$ $\left|C\right| =$ $12$, $D =$ $\{\frac{4}{C},$ $\frac{2}{T},$ $\frac{3}{A},$ $\frac{3}{G}\},$ $\left|D\right| =$ $12$ and $E =$ $\{\frac{3}{C},$ $\frac{2}{T},$ $\frac{3}{A},$ $\frac{4}{G}\},$ $\left|E\right| =$ $12$. Hence, $M^{*}J(AB) = 0.286$, $M^{*}G(AB) = 0.306$, ${M^{*}C(AB)} = 0.17$, ${M^{*}S(AB)} = 0.17$, $M^{*}J(AC) = 0.154$, $M^{*}G(AC) = 0.16$, ${M^{*} C(AC)} = 0.084$, ${M^{*} S(AC)} = 0.084$, $M^{*}J(AD) = 0.286$, $M^{*}G(AD) = 0.306$, ${M^{*}C(AD)} = 0.167$, ${M^{*}S(AD)} = 0.17$, $M^{*}J(AE) = 0.4$, $M^{*}G(AE) = 0.438$, $M^{*}C(AE) = 0.25$ and $M^{*}S(AE) = 0.25$. But, the balance of dissimilarity is $(A, A) = 0$, $(A, B) = 0.17$, $(A, C) = 0.08$, $(A, D) = 0.33$ and $(A, E) = 0.25$, according to the NCBI ^[44]. By matching the results of the association indices with the reports from the NCBI, it was found that the association indices $M^{*}C$ and $M^{*}S$ are the best.

6.   Conclusions and discussion

The complicated DNA research has become easier by using topology. Recently, many topologists found new methods to examine the mutations of DNA by using a combination of multiset topology and graph theory. Moreover, our presented results for repairing compatibility between the mathematical methods and biological solutions. In addition, we give a decision of mutation that is dependent on the metrics between two sequences of a gene and the topological structure derived from the relations. In the future, we can benefit from mutations by applying them end epidemics and in the fields of industry and agriculture. We have studied and identified mutations and showed how to make new ones, including how to fix mutations and apply Mathematica to construct models. Consequently, they are very significant in decision-making ^{[25,40,41,42]}. The introduced techniques are very useful in application because they pave the way for more topological applications for real-life problems. We also have an interesting application of our approaches to DNA sequences. The study of similarity between DNA sequences will be used to solve problems related to diseases and viruses, such as COVID-19 ^[16], which is an important example of mutations nowadays.

Use of AI tools declaration

The authors declare they have not used artificial intelligence tools in the creation of this article.

Acknowledgment

This work was supported by Researchers Supporting Project number (RSP2023R488), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that there are no conflicts of interest.

Appendix: MSC code