## 2 A Triplet Nucleotide Coding Scheme  
In this section, we present a new DNA coding scheme. We assume that a potential solution for an optimization problem consists of values for several variables, and is encoded in a single DNA strand. In the rest of this paper, we call an encoded potential solution an individual in a population (i.e., the portion of the solution space that is currently under exploration). A genetic algorithm will start with a random population, and search for global optimal solutions by generating new individuals from the individuals in the current population. These new individuals will be created by mimicking a cell reproduction process using a set of reproduction operators (to be presented in Section 3). When the algorithm terminates, the best fit individual will be decoded and presented as the optimal solution to the optimization problem. In biology, a DNA strand is a sequence of nucleotide bases (or simply nucleotides): Adenine (A), Guanine (G), Cytosine (C), and Thymine (T). A subsequence of three consecutive nucleotides is called a triplet codon (or simply a triplet) and represents an amino acid. During cell reproduction, a DNA strand is first translated into a sequence of RNAs, then into a sequence of an amino acid, and eventually into proteins. It is possible for different triplet codons to be translated into the same amino acid. In fact, the 64 unique triplet codons only correspond to 19 different amino acids. We define an amino acid-based coding scheme as follows. We map the 19 unique amino acids to integers 0 through 18, as shown in Table 1. For example, the triplet codon TTT is translated into amino acid Phe, which is mapped to 0, and TAG is translated into amino acid Stop, which is mapped to 9. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**First Nucleotide**} & \multicolumn{4}{c}{**Second Nucleotide**} & \multirow{2}{*}{**Third Nucleotide**} \\ \cline{2-2} \cline{4-5}  & **T** & **C** & **A** & **G** & \\ \hline T & Phe(0) & Ser(2) & Tyr(3) & Cys(4) & T \\ T & Phe(0) & Ser(2) & Tyr(3) & Cys(4) & C \\ T & [PERSON](1) & Ser(2) & Stop(9) & Stop(9) & A \\ T & [PERSON](1) & Ser(2) & Stop(9) & Try(9) & G \\ C & [PERSON](1) & Pro(5) & His(6) & Arg(8) & T \\ C & [PERSON](1) & Pro(5) & His(6) & Arg(8) & C \\ C & [PERSON](1) & Pro(5) & Gln(7) & Arg(8) & A \\ C & [PERSON](1) & Pro(5) & Gln(7) & Arg(8) & G \\ A & Ile(0) & Thr(11) & Asn(12) & Ser(2) & T \\ A & Ile(10) & Thr(11) & Asn(12) & Ser(2) & C \\ A & Met(9) & Thr(11) & Lys(13) & Arg(8) & A \\ A & Met(9) & Thr(11) & Lys(13) & Arg(8) & G \\ G & Val(14) & Ala(15) & Asp(16) & Gly(18) & T \\ G & Val(14) & Ala(15) & Abu(17) & Gly(18) & C \\ G & Val(14) & Ala(15) & Glu(17) & Gly(18) & A \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relationship between triplet codons and 19-ary integers. In our coding scheme, we also represent nucleotides A, G, C, and T numerically by 0, 1, 2, and 3, respectively. Thus, the triplet \(GTC\), representing amino acid Val, is represented numerically as 132. In general, an optimization problem with \(n\) variables can be defined as follows. \[\left\{ \begin{array}{l}\min f(x_{1},x_{2},...,x_{n})\\ x_{\min i}\leq x_{i}\leq x_{\max i},\,i=1,2,...,n\end{array} \right., \tag{1}\]  
where \(\mathbf{x}=(x_{1},x_{2},...,x_{n})\) is a vector of \(n\) decision or control variables, \(f(\mathbf{x})\) is the objective function to be minimized, and \([x_{\min i},x_{\max i}]\) is the value range (or domain) of variable \(x_{i}\). In a genetic algorithm, the objective function is often used to derive the fitness values that measure the quality of individuals. The individual with the best fitness value is chosen to be the best solution. To encode a possible solution, each variable \(x_{i}\) is represented by a base-4 integer of \(l\) digits and each possible solution (i.e., an individual) is represented by a sequence of \(n\) encoded variables. Therefore, the length of an individual is \(L=n\times l\). Here, \(l=3\times k\) and \(k\) represent the number of triplet codons per variable.