An automorphism in the LK_ITB5a H110F gene mutation

Sanchez et al. (2005) have shown the structure of vector space genetic code over the Galois field which relate to the physicochemical properties of the genetic code on proteins. From this vector space, an automorphism can be constructed to reflect the mutation process in the genetic code. This study investigates a type of transformation in the LK_ITB5a lipase gene mutation. The result of the study shows that there is a transformation matrix from the wild type gene lipase LK_ITB5a to mutant gene lipase LK_ITB5a H110F which is a diagonal matrix with non-zero determinant. This means that the transformation is an automorphism.


INTRODUCTION
The genetic code is a set of instructions in genes that express amino acids from a row of 3 bases on the RNA (ribonucleic acid) strand. Each of these 3 bases is called a codon. The genetic code is involved in the process of protein synthesis to translate strands of RNA codons into amino acids that form a protein molecule (Crick, 1968). This protein plays a role in the process of living and the growth of organisms. The genetic code is composed of a combination of 4 RNA bases, namely Adenine (A), Guanine (G), Cytosine (C) and Urasil (U). The genetic code has been presented in tables adjusted to the order of the second bases. One of the orders standard genetic code is presented in Table 1.
Genetic code bases can be presented in sets . Sanchez et al. (2005 proposed a vector space structure on the set after being inspired by binary representations made by previous researchers, such as Jimenez- Montano et al. (1996), Stambuk (2000) and Karasev and Stevanov (2001). The construction of the vector space begins with matching the set with . The matching is done based on the number of hydrogen bonds and the physicochemical properties of the bases. As a result, Sanchez et al. (2005) chose the representation , and . The match induces a group isomorphism between , and 〈 〉 . Furthermore, group 〈 〉 is a ring and Galois field . Hence, we can define an isomorphism such that also has a structure of Galois field GF(2^2 ) (Sanchez et al., 2005;.
Thereafter, the group of genetic code with 3 bases can The one letter symbol of amino acids.
be constructed as a product of group , namely . It can be checked that ( ) is also a commutative group. So, we can define the vector space genetic code over the Galois field . In addition, linear transformation can also be defined as a reflection of the genetic mutation process (Sanchez et al., 2005).
The mutation is a change in the genetic material of organisms that are inherited to the next generation. The mutation process is called mutagenesis, the mutated organism is called the wild type, and the mutation results are called mutants (Marwadewi, 2017).
Mutations can occur naturally and artificially. Natural mutations occur due to natural factors or errors of replication in the process of meiosis, artificial mutations are mutations due to factors from outside the body of the organism, such as ionization, enzymes or viruses, and the process of using chemicals. One of the artificial mutations that can be used is the polymerase chain reaction (PCR).
PCR is an enzymatic DNA replication technique without using organisms (Hasibuan, 2015;Ma'ruf, 2017). PCR can multiply DNA in a short time, therefore, the use of PCR is still the choice for DNA analysis in various fields of biochemistry and molecular biology. Ma'ruf (2017) used PCR to mutate the LK_ITB5a lipase gene to get certain characteristics of mutant that will be needed in the industry. This study will investigate whether the claims made by Sanchez et al. (2005) can also apply to every type of mutation, including artificial mutations.

VECTOR SPACE AND LINEAR TRANSFORMATION ON GENETIC CODE
The mathematical framework of this paper will just be the vector space and linear transformation on genetic code. First, the vector space built from the set RNA bases . Second, the vector space of genetic codes can be developed by the product of RNA bases vector space. Finally, we can construct the linear transformation to describe the mutational pathways on genetic code.

Vector space genetic code
Since is a bijective function, there is → such that for all we can define the sum "+" and product " " as: ( 2) (3) From the Equations (2) and (3), we have a Galois Field Genetic Code Bases . The operation result can be seen in Table 2. Based on the Galois Field of genetic code, we can define some vector spaces.

Definition 1 (Vector space of genetic code bases)
Commutative group is a vector space over Galois field with the sum and scalar product defined as follow: (4) for all . Furthermore, the base of is , so .

Definition 2 (Vector space genetic code)
Commutative group is a vector space over Galois field with the sum and scalar product defined as follow: for all . Furthermore, the standard base of is , so . The vector space genetic code over Galois field is called . Finally, this structure can be extended to vector space with genetic codes, so we have the vector space genetic code -dimension.

Definition 3 (Vector space of -dimension genetic codes over Galois Field )
Let be a product of groups of , that is:  (Crick, 1968). These results also suggests the strong connection between codon algebraic structure with the physicochemical property.

Automorphism on vector space of genetic code
The simplest mutation process is point mutation. A point mutation can be seen as the local endomorphism on the vector space of genetic code as in Definition 4 and Theorem 5 (Sanchez et al., 2005).

Definition 4 (Local endomorphism)
Let be a vector space over the Galois field , with . An endomorphism → will be called local endomorphism if there is and there are (Sanchez et al., 2005), such that: (9) and (10) If , then is local automorphism, endomorphism is diagonal local automorphism if and , for all . This means that is local endomorphism if one of the bases has changed, namely: where ∑ .

Theorem 1
For every single point mutation that change the codon of the wild type gene ( not the zero vector in vector space), by the codon of the mutant gene (Sanchez et al., 2005), there is: 1) At least a local endomorphism such that . 2) At least a local automorphism such that . 3) A unique diagonal automorphism such that if and only if the codon and of the wild type and mutant genes, repectively, are different of .

Proof (Sanchez et al.,2005)
Note that the purpose of the genetic code endomorphism study is to find out the mutation process in the genetic code so that we can understand the conditions before and after the mutation. Therefore, the selection of endomorphism is not arbitrary. To fulfill this goal, the endomorphism of chosen must have a reversal, that is, there is so that . Endomorphism which has this property is automorphism. Furthermore, the set of automorphisms is a group with composition operations.

Proposition 1. Let be a group of genetic code. Set
{ } is a group with composition ( " ).

Proposition 2.
Let be an endomorphism genetic code, and let be the matrix representation of . Then is automorphism if an only if .
Furthermore, a set also applies to the same thing, which is, for each there is so that , for all ( ) In other word, the representation matrix , with and . As a result, representation matrix is automorphism, if .
Let be a stabilizer subgroup that fix base position , that is, where is the subset of codon conserving the same base position . Note that preserves one base, while local endomorphism changes one base. As a result, every can be seen as a composition of 2 local endomorphism , so that . Hence, if are the matrix representation for and , respectively, then . If and is local diagonal automorphism, then so is .

AUTOMORPHISM ON LIPASE LK_ITB5a H110F GENE MUTATION
Lipase is one of the enzymes used in the industry. Generally, industries need lipases with certain characteristics. One example is lipases that are resistant to high pH needed for processing textile waste (Ma'ruf, 2017).
In this study, the lipase gene used was LK_ITB5a lipase which had a 99% identity with lipase Pseudomonas Stutzeri LipC (Ma'ruf, 2017). The LK_ITB5a lipase gene clone consists of 936 base pairs encoding 311 amino acids and 1 start codon. The LK_ITB5a lipase was mutated by the PCR approach through site-directed mutagenesis on the residual oxyanion hole H 110 substituted to F 110 . This mutation is done to see the effect of these mutations on the character of lipase. The results of the chemical analysis showed that the mutation on LK_ITB5a wild type into the LK_ITB5a H110F mutant caused a change in the interaction between the enzyme and the substrate so as to increase the enzyme specifications.
Next, an algebraic approach will be observed, which is by seeing the mutation process as a linear transformation in the vector space of genetic code, namely: 1. The lipase gene LK_ITB5a has 936 bases that encode 1 start codon and 311 amino acids. Then, we have , so that is a vector space -dimension over the Galois field . 2. Changes in the genetic code in the LK_ITB5a H110F mutant actually occur in codons 115 and 111 (because the first codon is the start codon, it does not encode amino acids), that is, the missense mutation H110F on codon 111 and silent mutation T114T on codon 115, then based on Theorem 1 there is a local endomorphism in the genetic code vector space. 3. Consider the missense mutation on 111 th codon, → . Based on the earlier mentioned points in the study, there is . This can also indicate the presence of local endomorphism , where and , so that . Let: be the matrix representation of and , respectively. Then, matrix representation of is It can be calculated that , so the endomorphism of is automorphism. More specifically, the endomorphism is a diagonal automorphism. 4. Consider the silent mutation on 115 th codon, → . Changes occur in only the 3 rd base, and also preserve the 1 st and 2 nd base, meaning there is local endomorphism , so that , with the matrix representation It can be calculated that , then endomorphism is a local diagonal automorphism. It also appears that . This mutation is most common (Sanchez et al., 2005;. Now, consider the mutation process from wild type gene LK_ITB5a to mutant gene LK_ITB5a H110F. Then there is → with matrix representation , where and , with

Since
, so that is an automorphism.

Conclusion
From the observations, it was found that the artificial mutation in the lipase LK_ITB5a H110F can also be seen as a linear transformation in the vector space of the genetic code. Particularly, the linear transformation in LK_ITB5a H110F is an authomorphism.