# Mutation Rate Model Used in the DNA VIEW Program
## 5 Mutation Models and Paternity Testing  
Let us denote unordered pairs \(P=\{p_{1},p_{2}\},M=\{m_{1},m_{2}\},D=\{d_{1},d_{2}\}\) of alleged paternal, maternal, and child's STR profile, respectively, (\(V=\{P,D,M\}\)). As usual in paternity testing, we consider two hypotheses:  
**Hypothesis 1** (\(H_{1}\)). : _Man P is a father of [PERSON] assuming that [PERSON] is a mother of [PERSON]._  
**Hypothesis 2** (\(H_{2}\)). : _A man different than [PERSON] is a father of [PERSON] assuming that [PERSON] is a mother of [PERSON]._  
Now, from the Bayesian formula we get  
\[P[H_{1}|V]=\frac{1}{1+\frac{P[V|H_{2}]}{P[V|H_{1}]}\frac{P[H_{2}]}{P[H_{1}]}}, \tag{27}\]  
and by putting a priori probabilities equal to 0.5 we obtain  
\[P[H_{1}|V]=\frac{1}{1+\frac{P[V|H_{2}]}{P[V|H_{1}]}}, \tag{28}\]  
(for more details of the computations see [10; 23; 31; 32]). However, the computed probability depends on the mutation model (we will write \(P_{i}[V|H_{1}],i=1,2,3,4\)). Thus, we compute  
\[\rho_{i,j}=|P_{i}[H_{1}|V]-P_{j}[H_{1}|V]|,1\leq i<j\leq 4. \tag{29}\]  
The frequency of each result \(V\) is a frequency of alleles \(P\) and \(M\) and from Mendel's law, the probability of achieving from such a pair of parents the child's profile \(D\) (it also depends on a mutation, but the differences are negligible). We consider two different situations:  
* Maternal mutation situation, i.e., when \(\{m_{1},m_{2}\}\cap\{d_{1},d_{2}\}=\emptyset\). * Without maternal mutation when \(\{m_{1},m_{2}\}\cap\{d_{1},d_{2}\}\
eq\emptyset\). We show the maximal values of \(\rho_{i,j}\) in Table 2 and the distribution of \(\rho_{i,j}\) in Figure 1 (the differences in the case without maternal mutation between all models and in the case when maternal mutation takes place between Models 1 and 2 were lower than 0.02, thus negligible). \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Maternal Mutation**} & \multicolumn{3}{c}{**No Maternal Mutation**} \\ \hline  & **Model 2** & **Model 3** & **Model 4** & **Model 2** & **Model 3** & **Model 4** \\ \hline Model 1 & 0.00003 & 0.9977 & 0.4237 & 0.00003 & 0.0005 & 0.0003 \\ Model 2 & & 0.9977 & 0.4237 & & 0.0005 & 0.0003 \\ Model 3 & & & 0.9985 & & & 0.0006 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximal difference of probability of paternity values between \(i\)-th and \(j\)-th model (\(\rho_{i,j}\)) in two cases: with maternal mutation and without it, at the DSS818 locus. For example, the maximal value 0.9985 was achieved for \(P=(15,15),D=(7,15),M=(14,14)\). In this case, we have the overall mutation rate \(\omega=0.001\), and the frequencies of alleles \(7,14\), and \(15\) are equal to \(0.002718,\ 0.009399,\ \text{and}\ 0.000566\), respectively. Thus, we have the Table 3.  
The above result follows from the fact that in the case of Model 3, allele 15 is significantly more likely to be a mutation of allele 14 than a transfer from a putative father. Finally, we verify some remarks on the inconsistency of theoretical equations and their practical application. Let \(P(p,s)\) be the probability that two alleles \(p\) and \(s\) are transferred from the same allele (they are brothers). Then, we have  
\[P(p,s)=\left\{ \begin{array}{ll}(1-\omega)^{2}*\
u_{s},&\text{if }p=s,\\ (1-(1-\omega)^{2})v_{s}v_{p},&\text{otherwise}\.\end{array} \right. \tag{30}\]  
but, on the other hand, we have  
\[P(p,s)=\sum_{t}(t\to p)(t\to s)v_{t}. \tag{31}\]  
In the formula (31), the assumed mutation model plays an important role, whereas in (30), only the overall mutation rate \(\omega\) is important. The maximal differences for locus \(D58818\) between the values computed using these formulas for different \(p\) and \(s\) are equal to \(0.000172,0.000164\), and \(0.000123\), and for the same \(p=s=12\), are equal to \(0.000197,0.000271\), and \(0.000271\) for Models 1, 2, and 3, respectively. Model 4 always gives an error of 0. All of them are small, smaller than the mutation rate. Which equation should be chosen by the investigator?