Response to: DNA identification by pedigree likelihood ratio accommodating population substructure and mutations- authors' reply.

Dear Editor, In their letter to the editor, Egeland et al. [1] criticize the mutation model used in our paper [2], and propose that our comments about the mutation model used by Dawid et al. [3] are not convincing, because we do not provide any data in support of our assertions. Their criticisms are primarily based on three premises: 1) that our mutation model, presented on page 5 of our paper [2], is mathematically incorrect, because our equation 8 does not define a proper probability distribution (that is, the probabilities do not add to 1); 2) that our mutation model allows for production of alleles of zero or negative repeat sizes, which are not meaningful; and 3) that the model used in the paper by Dawid et al [3] uses the relationship between mutational transition probabilities and allele frequency on the basis that allele frequencies are representative of the stationary distribution of a mutation process, and hence, in the absence of natural selection, is presumably applicable to the sequence tagged repeat (STR) loci used in DNA forensics. Each of these issues needs further discussion, and we thank the authors for giving us an opportunity to explain them further. First, the mutation model, explained by equation 8 of page 5 of our paper [2], clearly states that the geometric distribution for Pr (X = x) applies to ‘alleles to change by adding or subtracting an absolute number of x repeat units’. Hence, by definition x > 0, and as noted just after equation 8 ‘equal probabilities for gaining or losing repeats are assumed’, it is incorrect to multiply the geometric terms by a factor of 2, as Egeland et al. have done [1]. Following this logic, our equation 8 mathematically represents a valid probability distribution, because the total probability of mutation (that is, X ≠ 0) becomes μ, by summing the individual terms over all non-zero positive integer values of X. In addition, we are not the first to use such formulations of a mutation model. Estoup et al. [4] used exactly the same representation for the two-phase mutation model of Di Rienzo et al. [5] (see the information box 1 on page 1592 of the report by Estoup et al. [4]). Both of these papers provide observational and theoretical support for such a mutation model, applicable to STR loci (microsatellites, in their terminology). Second, it is true that this mutation model allows for production of zero or negative repeat sizes of alleles. This is also true for the simple stepwise mutation model (SMM [6]), in which, by successive single-step changes towards contraction of allele sizes, the allelic states (designated by repeat size) can eventually become zero or negative. There are alternative methods to minimize the effect of such biological absurdities. For example, allele-size constraints may be introduced to avoid continual unlimited expansions or contractions [7,8], although evidence for the presence of such constraints for STRs without any phenotypic effect is not clearly established [8]. By contrast, near equiprobable contraction and expansion, together with decreased probability of large multistep mutations, reduce the chance for reaching absurd allele sizes to almost negligible values (discussed below). An alternative method is to add allele-size nomenclatures denoted by b, for some arbitrary allele sizes a and b, defined by the smallest and largest sizes of alleles found at a locus in worldwide surveys (often called ‘below and above allelic ladder alleles’ [9-11]). The example chosen by Egeland et al. [1] also overemphasizes the possible occurrences of such unrealistic mutations. Incorrectly citing us, they used a mutation rate of 10 for the TH01 locus whereas in our paper, we only said that the STR loci used in forensic DNA analyses have mutation rates in the order of 10 to 10/locus/generation. Compilations of mutational * Correspondence: Ranajit.chakraborty@unthsc.edu Department of Forensic and Investigative Genetics, University of North Texas Health Science Center, Fort Worth, Texas 76107, USA Full list of author information is available at the end of the article Chakraborty et al. Investigative Genetics 2011, 2:8 http://www.investigativegenetics.com/content/2/1/8