# Mutation Rate Model Used in the DNA VIEW Program
## 3 Mutation Models  
Throughout this paper, the notation \((i\to j)\) means allele \(i\) was given by parents and arises as allele \(j\) in descendant. Although the maternal and the paternal mutations are generally different, we do not take this aspect into account, therefore we do not specify how the parent gives allele \(i\).  
Some assumptions (usually assumed) on equal chance of every allele to be transmitted are fulfilled too. Depending on whether mutation takes place or not, the following formula is used to describe the transmission of alleles,  
\[(i\to j)=\left\{\begin{array}{ll}1-\omega+\omega\omega_{i,i},&1\leq i=j\leq N,\\ \omega\omega_{i,j},&1\leq i,j\leq N,i\
eq j,\end{array}\right. \tag{3}\]  
where \(\omega_{i,j},1\leq i,j\leq N\) indicates how frequently \(i\) mutates into \(j\), provided that \(i\) mutates. The different mutation models are derived from a different evaluation of \(\omega_{i,j},1\leq i,j\leq N\). The value \(\omega_{i,i}\) means that there was mutation, but the number of gene repetition does not change (it is almost not possible to determine this value from observations). Historically, two models were considered:  
**Model 1.**  
\[\omega_{i,j}=\frac{1}{N}. \tag{4}\]  
**Model 2.**  
\[\omega_{i,j}=\left\{\begin{array}{ll}\frac{1}{N-1},&1\leq i,j\leq N,i\
eq j,\\ 0,&1\leq i=j\leq N.\end{array}\right. \tag{5}\]  
In the description of the DNA VIEW program (Mutation model implemented in DNA VIEW [20]), one can read the following.  
**Rule A.**  
Therefore, as a rule of thumb, I suggest assuming that  
50% of all mutations increase by one step  
50% decrease by one step  
5% increase by two steps  
5% decrease by two steps  
0.5% increase by three steps  
... etc.  
Never mind that these numbers add to more than 100%.  
This description leads us to the following model of mutation.  
**Model 3.**  
\[\omega_{i,j}=\left\{\begin{array}{ll}\frac{10^{[-|n_{i}-n_{j}|]}}{a_{i}},&1\leq i,j\leq N,i\
eq j,\\ 0,&1\leq i=j\leq N,\end{array}\right. \tag{6}\]  
where  
\[\alpha_{i}=\sum_{\begin{subarray}{c}1\leq j\leq N\\ j\
eq i\end{subarray}}10^{[-|n_{i}-n_{j}|]},\ \ \ 1\leq i\leq N. \tag{7}\]  
Previously, in the case of Variable Number of Tandem Repeat (VNTR) investigations, the most popular model of mutation was  
**Model 4.**  
\[\omega_{i,j}=\
u_{j},1\leq i,j\leq N. \tag{8}\]  
What assumptions should the mutation model satisfy? At first, we remark that a correctly defined mutation model should be such that  
\[\forall_{1\leq i\leq N}\qquad\sum_{j=1}^{N}(i\to j)=1, \tag{9}\]  
because the \(i\)-th allele must be transmitted \"to somebody\". On the other hand, the second condition deals with the allele frequencies in parent and child populations. At first, allele \(j\) has the frequencies \(\
u_{j}\), whereas after transmission, it has the frequency \(\sum_{i=1}^{N}(i\to j)\
u_{i}\). Thus, we have  
\[\forall_{1\leq j\leq N}\qquad\sum_{i=1}^{N}(i\to j)\
u_{i}=\
u_{j}. \tag{10}\]  
The last condition is a formulation of the Hardy-Weinberg hypothesis described in the Introduction.  
Using (3), these conditions can be formulated in an equivalent form:  
**Condition I.**  
\[\forall_{1\leq i\leq N}\qquad\sum_{j=1}^{N}\omega_{i,j}=1, \tag{11}\]  
**Condition II.**  
\[\forall_{1\leq j\leq N}\qquad\sum_{i=1}^{N}\omega_{i,j}\
u_{i}=\
u_{j}. \tag{12}\]  
Furthermore, it is easy to check that in some models (1 and 4) the computed total mutation is not equal to the observed \(\omega\). In Model 1 it is \((1-\frac{1}{N})\omega\) and for the \(i\)-th allele in Model 4 \((1-\
u_{i})\omega\) (instead of \(\omega\) in both cases). This is a small disturbance in the observed mutation rate.