# Mutation Rate Model Used in the DNA VIEW Program
## 1 Introduction  
Nowadays, DNA polymorphism is widely used in genetic expertise for fixing biological paternity and consanguinity of people. Results of DNA profiling are based on probabilistic and statistical interpretation. This interpretation leads to obtaining the probability of paternity or another relationship determined by Bayesian analysis (likelihood ratio). This probability depends on three factors:  
1. a priori probability (usually assumed to be equal to 0.5),
2. allele frequencies (main factor), and
3. mutation rate. The last factor, i.e., mutation rate, is usually small, but in some situations essential. From a viewpoint of population, mutation is the primary generator of genetic diversity, whereas drift tends to reduce the genetic diversity. Most of the human genome has a very low mutation rate: for a typical nucleotide site \(\omega=2\times 10^{-8}\) (the probability that a mutation will occur during gene transmission from a parent to a child). Short Tandem Repeat (STR) (also known as a microsatellite) mutation rates are even higher (approximately 1 or 2 mutations per locus per thousand generations). However, the key problem is that statistical analysis of the observed mutation cases is therefore almost impossible, because there is not a sufficient amount of observations (cf. [1, 2]). Some observations suggest taking the Stepwise Mutation Model (SMM; cf. [3, 4, 5]) in which a mutant allele has either \(k-1\) or \(k+1\) repeats, each with a probability of \(\frac{1}{2}\), where \(k\) is the current repeat number. The SMM is known to be false because the mutation rate increases with allele length and occasionally two-step mutations occur. In [6, 7, 8], other defects of the SMM are shown. Like the Mendel law, the Hardy-Weinberg hypothesis (sometimes called Hardy-Weinberg model/equilibrium/theorem/state of balance) is one of well known genetic laws. The Hardy-Weinberg model describes the genotype and allele frequencies in a non-evolving population. This model has seven basic assumptions (cf. [9, 10]):  
* organisms are diploid;
* only sexual reproduction occurs;
* generations are nonoverlapping;
* mating is random;
* population size is infinitely large;
* each sex has the same allele frequencies;
* there is no migration, gene flow, admixture, mutation, and selection. Despite these assumptions, as we show below, there exist mutation models (Model 4) in which the Hardy-Weinberg hypothesis holds. Given these assumptions, the population's genotypes and allele frequencies remain unchanged over successive generations, and the population is said to be in the Hardy-Weinberg equilibrium (cf. [11]). In practice, if we consider loci with \(\{A_{1},A_{2},\ldots A_{k}\}\) possible alleles and we observe the frequencies of a pair of alleles \((A_{i},A_{j})\) in the population \(P_{i,j},1\leq i\leq j\leq k\), then the good approximation (as the right-hand side values are estimated only from our database sample, and, as a consequence, these values in the whole population can slightly differ) of allele frequencies is  
\[p_{i}\approx\sum_{1\leq j\leq k,j\
eq i}P_{i,j}+2P_{i,i},\quad 1\leq i\leq k. \tag{1}\]  
The Hardy-Weinberg equilibrium can be stated as  
\[P_{i,j}\approx\left\{ \begin{array}{ll}p_{i}^{2},&\text{if $i=j$,}\\ 2p_{i}p_{j},&\text{otherwise.}\end{array} \right.\quad 1\leq i\leq j\leq k. \tag{2}\]  
The above approximation is usually (cf. [12] Chapter 3 and [13] Section 3.2) verified by [PERSON]'s \(\chi^{2}\) goodness-of-fit test (cf. [14, 15], here arises the problem of rare alleles whose frequencies must be bound together), [PERSON]'s exact test [16, 17, 18], or the likelihood ratio test or permutation test (known as the G-test, cf. [19]). Large deviations from the Hardy-Weinberg equilibrium are a consequence of errors in the STR amplification procedure, whereas the small ones follow from the departure from the above assumptions. In most genetic laboratories, the interpretation of forensic investigations is obtained as a result of computations of [PERSON]'s DNA VIEW computer program [20, 21].