# Mutation Rate Model Used in the DNA VIEW Program
## 4 Hardy-Weinberg Equilibrium
Mutation model 4 always satisfies both conditions._  
Proof of Corollary 2.: Let us consider the following matrices. \[A_{N}^{(1)}=\left[ \begin{array}{ccccc}\frac{1}{N}&\frac{1}{N}&\frac{1}{N}&...&\frac{1}{N}\\ \frac{1}{N}&\frac{1}{N}&\frac{1}{N}&...&\frac{1}{N}\\ \frac{1}{N}&\frac{1}{N}&\frac{1}{N}&...&\frac{1}{N}\\ \frac{1}{N}&\frac{1}{N}&...&\frac{1}{N}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{1}{N}&\frac{1}{N}&\frac{1}{N}&...&\frac{1}{N}\\ \end{array} \right],\ \ A_{N}^{(2)}=\left[ \begin{array}{ccccc}0&\frac{1}{N-1}&\frac{1}{N-1 }&...&\frac{1}{N-1}\\ \frac{1}{N-1}&0&\frac{1}{N-1}&...&\frac{1}{N-1}\\ \frac{1}{N-1}&\frac{1}{N-1}&0&...&\frac{1}{N-1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{1}{N-1}&\frac{1}{N-1}&\frac{1}{N-1}&...&0\\ \end{array} \right], \tag{22}\]  
and  
\[A_{N}^{(3)}=\left[ \begin{array}{ccccc}0&\frac{10^{[-1]n_{1}-n_{2}]}}{a_{n_{ 1}}}&\frac{10^{[-1]n_{1}-n_{3}]}}{a_{n_{1}}}&...&\frac{10^{[-1]n_{1}-n_{N}]}}{ a_{n_{1}}}\\ \frac{10^{[-1]n_{2}-n_{1}]}}{a_{n_{2}}}&0&\frac{10^{[-1]n_{2}-n_{3}]}}{a_{n_{2}}} &...&\frac{10^{[-1]n_{2}-n_{N}]}}{a_{n_{2}}}\\ \frac{10^{[-1]n_{3}-n_{1}]}}{a_{n_{3}}}&\frac{10^{[-1]n_{3}-n_{2}]}}{a_{n_{3}}} &0&...&\frac{10^{[-1]n_{3}-n_{N}]}}{a_{n_{3}}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{10^{[-1]n_{N}-n_{1}]}}{a_{n_{N}}}&\frac{10^{[-1]n_{N}-n_{2}]}}{a_{n_{N}}} &\frac{10^{[-1]n_{N}-n_{3}]}}{a_{n_{N}}}&...&0\\ \end{array} \right]. \tag{23}\]  
Now, if \(v_{o}\) denotes the vector of allele frequencies of parents' population, then Condition II (Hardy-Weinberg hypothesis) can be rewritten as  
\[(A_{N}^{(k)}-I_{N})v_{o}=0,\ \ k=1,2,3, \tag{24}\]  
where \(I_{N}\) denotes the \(N\times N\) unit matrix. From Corollary 1 we have \(rank(A_{N}^{(k)}-I_{N})=N-1,k=1,2,3\). Thus, the subspace of solutions of \(A_{N}^{(k)}x=x,k=1,2,3\) is one-dimensional. On the other hand, it is easy to see that the vector \(\vartheta_{N}=[\frac{1}{N},\frac{1}{N},...,\frac{1}{N}]^{T}\) satisfies \(A_{N}^{(k)}\vartheta_{N}=\vartheta_{N},k=1,2,3\), thus it is a unique solution (among the vectors with non-negative coordinates such that \(||x||_{I_{1}}=1\)). **Theorem 2**. : _Let us define the sequence of vectors \(\{v_{k},k\geq 1\}\) by_  
\[v_{k+1}=\big{(}(1-\omega)I_{N}+\omega A_{N}^{(i)}\big{)}v_{k}=T_{i}v_{k},\ \ \ k=0,1,2,...,\ \ \ i=1,2,3, \tag{25}\]  
_where \(v_{o}\) denotes the vector of starting allele frequencies in some loci, \(N\) is the number of alleles, and \(i=1,2,3\) is the selected model of mutations (\(v_{k}\) denotes the frequencies of alleles in the \(k\)-th population applying the \(i\)-th model of mutation). Then,_  
\[\lim_{k\rightarrow\infty}T_{i}^{k}v_{o}=\lim_{k\rightarrow\infty}\underbrace {(T_{i}\circ T_{i}\circ\ldots\circ T_{j})v_{o}=\lim_{k\rightarrow\infty}v_{k}= \vartheta_{N},\ \ \ i=1,2,3. \tag{26}\]  
Proof of Theorem 2.: Let us consider the Banach space \(\Re^{N}\) with the norm \(||x||_{I_{1}}=\sum_{i=1}^{N}|x_{i}|\). Furthermore, let \(K=\{x\in\Re^{N}:||x||_{I_{1}}=1,0\leq x_{i}\leq 1,i=1,2,3,...,N\}\) be a compact, separable subset of the space \(\Re^{N}\). Obviously, the fixed point of the mapping \(T_{i}\) on \(K\) is the same as the fixed point ofand from Corollary 3 this point is \(\sigma_{N}\). The mapping \(T_{i}\) is not a contraction because \(||T_{1}||=||A_{N}^{(i)}||=1\), as the matrix \(T_{i}\) consists of non-negative numbers such that their sums are equal to 1 in each row and each column. The norm is achieved for an arbitrary vector \(x\) with \(||x||_{I_{1}}=1\) and all non-negative or all non-positive coordinates. Thus, we cannot apply the Banach Contraction Theorem. However, Theorem 9.4 (and Corollary 9.1) in [28] considers the matrix such as \(T_{i}\). As a conclusion we get a thesis. More information about the determinants and algebra of matrices can be found in [29; 30].