Eigenvectors determination of the ribosome dynamics model during mRNA translation using the Kleene Star algorithm

Eigenvalues and eigenvectors in max-plus algebra have the same important role as eigenvalues and eigenvectors in conventional algebra. In max-plus algebra, eigenvalues and eigenvectors are useful for knowing dynamics of the system such as in train system scheduling, scheduling production systems and scheduling learning activities in moving classes. In the translation of proteins in which the ribosome move uni-directionally along the mRNA strand to recruit the amino acids that make up the protein, eigenvalues and eigenvectors are used to calculate protein production rates and density of ribosomes on the mRNA. Based on this, it is important to examine the eigenvalues and eigenvectors in the process of protein translation. In this paper an eigenvector formula is given for a ribosome dynamics during mRNA translation by using the Kleene star algorithm in which the resulting eigenvector formula is simpler and easier to apply to the system than that introduced elsewhere. This paper also discusses the properties of the matrix Bλ⊗n of model. Among the important properties, it always has the same elements in the first column for n = 1, 2,… if the eigenvalue is the time of initiation, λ = τin, and the column is the eigenvector of the model corresponding to λ.


Introduction
Max-plus algebra is widely used to model time events discrete system. A typical application of discrete events system are production lines, where every machine must wait with a starting a new operation until the operations on other machines are completed as in the scheduling, transporting, manufacturing, queuing, and protein production in cell. One problem that often appears in the maxplus algebra is the eigen problem. The eigen problem for max-plus matrices describes the steady state of the system. A more details examples of the eigen problems can be seen in [4], [5], [7], [9], [10], [11].
The eigenvalues and eigenvectors in a max-plus model can be used to get an overview of the system dynamics such as in train system scheduling, scheduling production systems, and scheduling lesson activities on moving classes. In the translation stage of cellular protein production, in which the ribosomes moves uni-directionally along an mRNA strand to building amino acid chains, the eigenvalues and eigenvectors are useful to calculate the protein production rate and density of ribosomes on the mRNA. Based on this, it is important to examine the eigenvalues and eigenvectors of the model. In the previous work, the authors in [3] provided a formula for determining the eigenvalues and eigenvectors of their model [3]. In this paper we will discuss a simpler formula that is easier to understand the process of determining the eigenvectors of their model. This formula is obtained by a different approach compared to the work in [3].

Model and Method
The details derivation of the model are given in [3]. In this section we only present a brief theoretical background required in the subsequent sections, such as mRNA translation, max-plus algebra, eigen problem in max-plus algebra and matrices of the ribosome dynamics during mRNA translation.

Messenger RNA translation
mRNA translation is one of the steps in protein production in cells [1]. The mRNAs are single strands of nucleotides, grouped in triplets called codons, holds the code for a specific chain of amino acids that makes up a protein. The translation is performed by molecular machines called ribosomes, which scan along the mRNA adding amino acids to a growing chain which will become the protein. In this paper we will examine the eigenvalues and eigenvectors of the matrix B in (1) which is a max-plus model of ribosome dynamics during mRNA translation introduced by [3]. See [3] for the details. 1 1

Max-plus algebra
Bacelli et al. in [2] defines the max-plus algebra and denoted by ℛ ௫ which is defined as the set ℝ ௫ = ℝ ∪ −∞ , with ℝ is real numbers extended by an infinite element ߝ = −∞ and operation ⊕ , where ܽ, ܾ ∈ ℝ ୫ୟ୶ . It is easy to show that both operation ⊕ and ⊗ are associative and commutative. The zero and unit element in max-plus algebra are ߝ = −∞ and ݁ = 0, respectively. We define max-plus power in the natural way We can write ܽ ⊗ in conventional algebra as ݇ × ܽ. We also note that the ⊗ operator has an inverse which can be expressed as a negative power but there is no inverse of the ⊕ operator.
Vectors and matrices can also be constructed. We denoted by ܽ the i-jth component of the matrix ‫.ܣ‬ For matrices ‫ܣ‬ ∈ ℝ ௫ × , ‫ܤ‬ ∈ ℝ ௫ × , ‫ܥ‬ ∈ ℝ ௫ × the sum and product of matrices are then defined as For square matrix ‫ܣ‬ ∈ ℝ ௫ × , similar to scalar in max-plus algebra, we denote with n E is the identity matrix in ℝ ௫ × . Furthermore, those relation can be written in the which is called Kleene star. Therefore, Eigen problems are common problem in mathematics especially in linear algebra. In max-plus algebra, similar to linear algebra, eigen problems are formulated as A λ is called eigenvector. This problem is treated by several authors who can be such as in [4], [5], [10], [11]. The graph associated with matrix ‫ܣ‬ ∈ ℝ ௫ × , A with value ij a ε ≠ , then there is an arc j i → ሺfrom j to iሻ with weight ij a . A series of one or more arcs between two nodes i and j is called a path i j → . A path i i → is called a circuit and denoted γ . A graph is said to be strongly connected if for any two different nodes there is a path between them and a matrix A is said to be irreducible if the corresponding graph ( ) G A is strongly connected. The circuit weight w γ of a circuit γ is defined as the sum of the weight of all arcs in that circuit and the circuit length l γ as the number of arcs in the circuit. The mean circuit weight is then defined as for any λ . For the analysis of the subsequent section we need the following theorems: × be a matrix that has a maximal mean circuit weight of circuits in ( ) G A equal to e then A + exists and is defined by 2 1 ...
Based on Theorem 1 it is given that , and according to (7) it is obtained that We do not prove the theorems. Interested readers can consult the respective references.

Result and Discussion
In this section, we will examine how to determine the eigenvector of the matrix B in (1). The eigenvalue of the matrix B can be found from the correspomding graph ( ) G B . Fig. 1 show the example of graph ( ) G B with 3 n = .
b. If c. If The next discussion is purported to calculate the eigenvectors which different to the method in [3]. It is done by using the Kleene star algorithm. We identify the eigenvectors into three cases: Several other authors have reviewed this method such as [8]. By using this method we find the following theorem regarding the properties of the matrix n B λ ⊗ of (1) as follows: and so on. It can be seen that all the first columns of the matrix n B λ ⊗ have elements that are always the same for ݊ = 1,2,3, … ii.
In the next section will examine the eigenvectors of the matrix (1) by the Kleene star algorithm .
By continuing this step for the matrix ( ) ( )

Case (iii):
In this case we assume that there are two places in the bulk of the mRNA, with waiting times equals to τ . That is, there exist integers p and q such that ,