Inferring the Weight of Evidence from two-person mixtures when explored by Y-chromosome haplotypes

In recent decades, the improvement in the sensitivity of commercial kits used for the amplification of genetic markers in forensic work, together with the increase in the complexity of the cases dealt with, has led to an increase in the number of mixed traces submitted to the forensic expert for analysis as a model of evidence in molecular investigations aimed at criminal identification. The interpretation of mixed traces analyzed by autosomal markers has been an intensively studied field, and at least two computational methods are now available to the expert: qualitative (semi-continuous) and quantitative Bayesian (continuous). A good assortment of application software inspired by these principles is also available today, with products that can be purchased commercially or acquired as free shareware (for example, LRmix Studio, EuroForMix, CaseSolver, etc.). Some of these software are surveyed and, to some extent, approved by the International Society of Forensic Genetics. However, what still makes the theoretical and application landscape on the subject of mixed traces unsatisfactory is the almost total absence of principles and analysis methods that can be applied to the field of markers located on the Y chromosome (Y-Chr). In this context, the main problems that make theoretical elaboration and interpretative progress complex are as follows: • The Y-Chr is theoretically inherited en bloc, and the presence of haplotypes makes extrapolating population frequency data that can be integrated into individualization calculations challenging; • A unique haplotype frequency calculation model does not yet exist; • The approach to the identification of all possible combinations resulting from a mixed profile has to deal with the construction of pairs, triplets, quadruplets, etc. (in the case of mixtures with two, three, and four contributors, respectively) of haplotypes (and not of single genotypes); • Many haplotypes generated by an albeit correct haplotype combinatorial process may never have been detected in the population surveys carried out by the laboratories; therefore, their existence may legitimately be questioned; • Solutions for the deconvolution of Y-STRs (Short Tandem Repeats) mixtures are still practically non-existent. This work aims to illustrate a computational procedure to extract and evaluate the Power of Evidence (WoE) in biological mixtures corresponding to the two contributors' model, studied employing a genetic marker protocol on the Y-Chr. In this study, the Y-STRs mixtures are represented with genotype permutation universal matrices, which include several columns equivalent to the number of putative contributors and many rows/states comparable to the number of different permutations incorporating the evidence. After the matrices' construction, a calculation method was proposed based on the assumption of measurable quantities, gene frequencies assumed to be all available, and no drop-out. The three independent indices (Peak Height ratio, Mix Ratio, and χ2) were modelled, and a specific description of how they were calculated is given in the manuscript. The theoretical development of this work was described following the two contributors – ten loci model; practical calculation examples helpful in testing the effectiveness of the method developed were then described using the two contributors – seven loci model, due to the low data processing capacity of the personal computers used in this work. The spreadsheets developed in Microsoft Excel, relating to matrix construction and calculation of the Likelihood Ratio, are user-friendly and accessible for reference upon explicit request to the Authors.