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IPATIMUP, Institute of Molecular Pathology and Immunology of the University of Porto, PortugalFaculty of Sciences of the University of Porto, PortugalCMUP, Centre of Mathematics of the University of Porto, Porto, Portugal
In paternity testing the genetic profiles of the individuals are used to compare the relative likelihoods of the alleged father and the child being related as father/offspring against, usually, being unrelated.
In the great majority of the cases, analyses with the widely used sets of short tandem repeat markers (STRs) provide powerful statistical evidence favouring one of the alternative hypotheses. Nevertheless, there are situations where the final statistical result is ambiguous, mostly because the alleged father shows incompatible genotypes at a few loci along with a very high paternity index in the remaining systems. In these cases, the possibility that the alleged father is actually a close relative of the real one (son, father or brother) can reasonably be raised.
In such cases, when the statistical evidence obtained is considered as insufficient, the common practice is to extend the set of analysed markers. In this context, many authors have suggested that bi-allelic markers, such as single nucleotide (SNP) or insertion/deletion (Indel) polymorphisms, are markers of choice, as they are incomparably less prone to mutation than STRs.
In this work we address the soundness of this claim and the consequences of this strategy, analyzing the a priori odds both for (a) expected number of Mendelian incompatibilities, and (b) expected values for the final likelihood ratios. Moreover, one hundred real pairs of second degree relatives, typed for two sets of markers: 15 STRs plus 38 Indels, were used to simulate paternity testing. Our data show that, for the number of markers commonly considered, the results from an extended battery of SNPs or Indels should be interpreted with caution when relatives are possibly involved.
In the framework of forensic kinship evaluation, paternity is the most common analysis. In this case the genetic profiles of two individuals (or three, if the mother is available for testing) are used to compare the relative likelihoods of being related as father/offspring against, usually, being unrelated. In this context, it is commonly accepted that the unlinked autosomal Short Tandem Repeat markers (STRs) are the preferred tool for kinship evaluation due to their high polymorphism, as they usually provide very powerful statistics favouring one of the alternative hypotheses [
]. It has been recommended that the weight of the evidence should be based on the calculation of likelihood ratios (LRs), generally considering unlinked markers and, in consequence, applying the so-called ‘product rule’ to the LRs obtained for each marker [
However, experts are quite often faced with situations where the final LR value is ambiguous, mostly because the alleged father shows incompatibilities at a small number of loci along with a high paternity index in the remaining (compatible) systems. In these cases, the possibility that the alleged father is a close relative of the real one can reasonably be raised, viz., the alleged father is a brother of the true one (the tested man is, then, uncle of the child), or, as it occurs frequently in immigration cases [
], a second degree relative is presented as the alleged father. In this work it will be globally considered the situations where the alleged father is a brother, a son or the father of the true father because the resulting pedigrees linking the alleged father and the child (uncle–nephew/niece, half-siblings and grandfather–grandchild, respectively) belong to the same autosomal kinship class [
], i.e. the pedigrees are indistinguishable when using unlinked autosomal markers.
Genetic relatedness is always studied with basis on the root concept of identity-by-descent; two individuals are said to be related if some allele of one can be identical-by-descent (or IBD) to some allele of the other [
]. In this context, it should be clear that any kinship (other than unrelated) between alleged father and child raises the probability of sharing identical alleles; closer and distant kinships implying, respectively, greater and smaller probabilities of sharing IBD alleles. Particularly for the genealogies uncle–nephew/niece, half-siblings or grandfather–grandchild, there is one half of probability of the individuals sharing one pair of IBD alleles, in addition to the possibility of the individuals sharing identical alleles by chance (i.e. alleles identical-by-state but not identical-by-descent).
In any case, when the statistical evidence obtained is considered as insufficient, the common practice is to enlarge the set of analysed markers. Many authors have suggested that the bi-allelic markers single nucleotide polymorphisms (SNPs) and insertion–deletion polymorphisms (Indels) are markers of choice to extend such a battery, as they are much less prone to mutation than STRs (see e.g. [
]). In this paper we will address the soundness of this claim, analysing algebraically and with real and simulated cases the consequences of this strategy, discussing the impact of the use of SNP/Indel (or, broadly speaking, bi-allelic) markers on paternity testing, with a special emphasis on the issue of a possible kinship between the alleged and the true father, as in cases where the alleged father and the child are second-degree relatives. The results will be dissected at the light of the expected behaviour of the bi-allelic markers in terms of LR distributions and number and type of incompatibilities.
It is expected that an increase of the background level of relatedness of the population will raise the probability of sharing identical alleles; in this work, however, we will disregard such parameter, following the recommendations given by Gjertson et al. [
]. Indeed, for standard populations, the typical levels of inbreeding are very small, not influencing in a significant degree the measurement of the weight of the evidence in paternity analyses (see [
Henceforth both SNPs and Indels will be globally referred to as bi-allelic markers (BAMs), although they can present more than two alleles in a population.
The theoretical calculations were performed for specific STR batteries and for simulated sets of BAMs assuming independence. The STR batteries analysed were: (a) the 15 markers included in the commercial kit Identifiler®, Applied Biosystems; (b) the 17 markers resulting from the combination of both Identifiler®, Applied Biosystems, and Powerplex 16, Promega Corp. kits; (c) the previous 17 STRs and four extra STR markers developed and validated in-house: CD4, F13, FES and MBPB [
]. The database considered for the STR allele frequencies was that used in the Laboratory of Parentage Tests and Genetic ID of IPATIMUP Diagnostics, corresponding to a random sample of unrelated residents in North Portugal.
It has been considered that the observation of fully codominant Mendelian incompatibilities under the assumption of paternity have two alternative explanations: (a) a silent allele is present (Landsteiner's 2nd rule, or 2nd order incompatibility), or (b) a de novo mutation occurred (Landsteiner's 1st rule, or 1st order incompatibility). Indeed, when the individuals alleged father (AF) and child (C) are apparently homozygous for different alleles (both in duos and trios), under the assumption of parenthood such configuration of genetic types can be explained considering the presence of a silent allele, and these are the so-called 2nd order incompatibilities or by Landsteiner's 2nd rule. On the other hand, if the two alleles of C are both absent from AF (both in duos and trios cases) or C shows an allele absent from both AF and mother, any incompatibility is only explainable considering that a mutation had occurred – 1st order incompatibilities or by Landsteiner's 1st rule. To incorporate such incompatibilities in the calculations, both mutation rates and frequencies of silent alleles were considered. For theoretical calculations, mutation rate values of 10−8 and 10−9 for, respectively, SNPs and Indels [
] were considered. Moreover, and since there is no reason to assume that silent alleles have different frequencies in STRs and in BAMs as they both rely on PCR based genotyping, the same frequency was considered in both cases: 0.001, the lower bound of the interval (0.001–0.005) reported in the literature [
The paternity testing was computed invoking mutations and silent alleles only when they were required for making compatible the genetic types under the assumption of parenthood (for all the analysed markers the estimated frequency of the silent allele was smaller than 0.005). When mutations were required to be considered, no distinction was made depending on the sex of individuals [
] and mutation rates were considered depending on the number of mutational steps, assuming equally likely the addition or the subtraction of one repeat.
Hereupon our approach to the problem followed the below-mentioned design.
First we calculated the expected probabilities of finding less than three Mendelian incompatibilities (which is usually considered to be problematic, likely leading to ‘difficult cases’) between (a) two unrelated individuals and (b) two 2nd degree relatives, in trios and duos, and for the different batteries of STRs previously mentioned.
The probability of finding none or one Mendelian incompatibility was then performed for different simulated sets of BAMs, but only under the assumption of second degree relatedness, a major fraction of the ‘difficult cases’ for which complementary markers are asked to contribute.
The distribution of the expected LR for each marker, in duos and trios, for both compatible and incompatible configurations of phenotypes will also be presented, as well as the distribution of the expected final LR values for simulated sets of 30, 50 and 100 BAMs in duos, in both the absence of incompatibilities and when exactly one incompatibility was found.
All the considered algebraic expressions are presented as supplementary material. The algorithms for computing the expected LR for the sets of BAMs were performed using MATLAB 7.10.0 (R2010a) software.
Finally, some examples of real cases were performed and 100 true uncle–nephew/niece and grandfather–grandchild unrelated duos were analysed with 15 STRs (commercial kit Identifiler®, Applied Biosystems) and 38 Indels (Indel-plex described by Pereira et al. [
3.1 Expected frequencies of ‘difficult cases’ using standard STR batteries
In Table 1, Table 2 are displayed the probabilities of finding none, one or two Mendelian incompatibilities for different sets of STRs in two circumstances: the alleged father and child are 2nd degree relatives or unrelated. We specify the probabilities for finding one and two incompatibilities for being those that commonly lead to ambiguous LR values, which does not mean that, depending on the analysed systems, a greater number of incompatibilities have associated a conclusive LR result (see Section 3.3).
Table 1A priori probabilities of finding none, one or two Mendelian incompatibilities in a duo paternity test, with alleged father and child being second degree relatives or unrelated, using batteries of: (a) 15, (b) 17 or (c) 21 STRs.
Table 2A priori probabilities of finding none, one or two Mendelian incompatibilities in a trio paternity test, with alleged father and child being second degree relatives or unrelated, using batteries of (a) 15, (b) 17 or (c) 21 STRs.
From these results we infer that, specifically for this database, if using the 15 STR set approximately 2.5% of duos will be expected to show no incompatibilities when a second degree relative is presented as alleged father (Table 1). The corresponding probability in trios drops to a much more tolerable figure of ≈0.4% (Table 2). On the other hand, 21 STRs would be required to expect zero incompatibilities in duos in less than 1% (≈0.7%) of the paternity cases, when the putative father is a second degree relative of the child.
Moreover, the expected probability of finding less than three genetic incompatibilities (which most likely will lead to inconclusive LR values) when a second degree relative is tested as alleged father vary from, approximately, 36% in the most adverse scenario (duos and 15 STRs – Table 1) to 3% in the most favourable (trios and 21 STRs – Table 2).
3.2 Extending genotyping to BAMs: expected probabilities
3.2.1 When no incompatibilities are found
Considering different sets of BAMs, for both different number of markers and allele frequencies, the expected chance of finding no incompatibilities between a pair of 2nd degree relatives is presented in Table 3, both in duo and trio cases. Mathematical procedure is described in supplementary material (see Section 1.1.1).
Table 3Probability of finding no Mendelian incompatibilities among two second degree relatives, considering duos and trios, and different batteries of bi-allelic markers.
From Table 3 we want to highlight that: (a) for a set of 30 BAMs, even with equally frequent alleles, it is expected that ≈14.4% of duos and ≈1% of trios involving 2nd degree relatives do not exhibit any incompatibility; (b) for duos a battery of at least 100 BAMs with equally frequent alleles is required to lower this figure to a value below 1%; (c) for trios 50 BAMs are sufficient to obtain a probability of less than 0.1%.
The expected LR for a BAM where individuals show compatible types, both for duos and trios, was performed considering the probabilities of two second degree relatives exhibiting each one of the possible compatible genetic types. The obtained results are depicted in Table 4 and the mathematical procedure is described in supplementary material (Section 1.1.2).
Table 4Expected LR for a compatible BAM in a paternity test where the alleged father is 2nd degree relative of the child.
From Table 4 it can be highlighted that, expectedly, compatible BAMs will reinforce the hypothesis of paternity, the evidence being stronger when a trio is analysed. Moreover, note that in duos the unbalance of the allele frequencies carries a decrease of the expected LR of the marker. On the other hand, in trios, the unbalance of the allelic frequencies leads to an increase of the expected LR value.
A priori calculations for the expected final LR values when, using different sets of BAMs, no incompatibilities are found in paternity testing for duos with alleged father 2nd degree relative of the child were also performed. Results are presented in Fig. 1 and in Table S3 in supplementary material (Section 1.1.3).
Fig. 1Order of magnitude (base 10 log scale) of the expected final values of paternity testing LR for duos with 2nd degree relatives and without incompatibilities. Simulated sets of 30, 50 and 100 BAMs are considered, either each marker with equally frequent alleles or with frequencies 0.4/0.6.
From these data it can be concluded that, generally, in the absence of incompatibilities the hypothesis of paternity is favoured. In such cases, for sets of 30, 50 and 100 BAMs with equally frequent alleles the modal LR order of magnitude is 1 (expectedly for ≈48% of the cases: 10 ≤ LR < 100), 3 (expectedly for ≈45% of the cases: 103 ≤ LR < 104) and 6 (expectedly for ≈34% of the cases: 106 ≤ LR < 107), respectively.
Therefore, if the alleged father is a close relative (brother, son or father) of the real one and that suspicion can be raised after analyzing a first set of STRs, when no incompatibilities are found in a complementary set of BAMs the additional results will likely complement those previously obtained with STRs apparently supporting a (wrong) conclusion of paternity. Note that, combining the data from Table 1, Table 2, Table 3, for, e.g. the first mentioned kit with 15 STRs and 30 BAMs with equally frequent alleles, it is expected that two 2nd degree relatives show one incompatibility in the STR set and none in the Indel set in ≈1.5% of the cases for duos and in ≈0.3% of the cases for trios.
3.2.2 When exactly one incompatibility is found
At this point it should be highlighted that, in paternity testing with BAMs, 1st order incompatibilities are only possible when the mother is available for testing (when she and the alleged father are both apparent homozygotes for the same allele and the child is heterozygote). Indeed, only 2nd order incompatibilities are possible in duos (when the alleged father and the child are apparent “opposite” homozygotes). Thus, the residual mutation rates of SNPs and Indels are only relevant when trios are analysed; for duos all the incompatibilities can be explained by the much more likely presence of a silent allele. The probabilities of finding exactly one incompatibility for different sets of BAMs in duos and trios are presented in Table 5, assuming, as before, that the alleged father and child are 2nd degree relatives. Mathematical procedure is presented in supplementary material (Section 1.2.1).
Table 5Probability of finding exactly one Mendelian incompatibility among two second degree relatives, considering duos and trios, and different batteries of BAMs.
In the same way as for compatibilities, the expected LR for an incompatible BAM, for duos and trios with relatives and (in trios) for both type of incompatibilities, was performed. The results are presented in Table 6. The most parsimonious arguments were used and, therefore, the presence of a silent allele was considered to explain, under the assumption of paternity, 2nd order incompatibilities. For 1st order incompatibilities, the mutation rate considered was that estimated for SNPs: 10−8 [
Table 6Expected LR for one incompatible BAM in a paternity test where the alleged father is 2nd degree relative of the child, for a frequency of the silent allele of 10−3 and a mutation rate of 10−8.
From Table 6 it can be seen that, expectedly, the evidence against paternity is approximately 105 times stronger when a 1st order incompatibility is found. Indeed, for SNPs and Indels, 1st order incompatibilities are much more effective to discard false fathers than 2nd order ones.
Finally, considering duos and different sets of BAMs, the expected LRs for paternity testing with relatives was performed assuming that exactly one (2nd order) incompatibility was found. The results are depicted in Fig. 2 and in Table S4 in supplementary material (Section 1.2.3).
Fig. 2Order of magnitude (base 10 log scale) of the expected values of LR for duos with 2nd degree relatives and one (2nd order) incompatibility. Simulated sets of 30, 50 and 100 BAMs were considered, either each marker with equally frequent alleles or with frequencies 0.4/0.6.
From the results obtained for markers with equally frequent alleles we want to stress that, expectedly, when one (2nd order) incompatibility is found: (a) with a set of 30 markers ≈21% of the cases will favour paternity; (b) with a set of 50 markers ≈79% of the cases will favour paternity, ≈5% with LR order of magnitude of 2; (c) with a set of 100 markers it is almost certain that the LR will favour paternity, ≈32% with LR order of magnitude between 4 and 6.
Indeed, depending on the number of markers, the analysis of BAMs can be insufficient to obtain a final LR value against paternity.
3.3 Some examples of real cases
A total of 100 real cases of uncle–nephew/niece and grandfather–grandchild duos were typed for 15 STRs and 38 Indels. For each duo a paternity testing was simulated, comparing the probability of the observations under the hypothesis of paternity with the probability of the same observations assuming, alternatively, the individuals as unrelated. A final LR value comprised between 10−4 and 104 was considered as inconclusive.
Considering just the STR analysis, 33 cases revealed to be inconclusive, 5 of which with a LR favouring paternity. In the inconclusive cases, the LR varied between 2.1E−4 and 1584, and the number of incompatibilities varied between 1 and 3 (indeed, for two duos, 3 incompatibilities were not sufficient to obtain a LR value capable of discarding paternity).
For the Indel-plex analysis, 17 duos did not show any incompatibility (in these cases: 7 ≤ LR ≤ 751) and 37 showed one incompatibility, 10 of which with final LR favouring paternity (1.1 ≤ LR ≤ 15.5).
Combining both STR and Indel results, 25 cases remained inconclusive, 5 of which with LRs favouring paternity: 5.7 ≤ LR ≤ 3389.6.
In paternity testing the use of STRs and a battery of SNPs or Indels can be helpful to exclude false fathers but should be taken with caution for inclusion, particularly when, after analyzing a first set of STRs, the suspicion that the alleged father is a close relative of the real one cannot be ruled out. Indeed, the low polymorphism of bi-allelic markers leads to a large increase of the probability of two individuals sharing identical alleles by chance (i.e. not by descendent). When, additionally, the individuals are kin related (by other kinship than paternity) that probability is (critically) high, leading to a significant proportion of cases where, expectedly, the individuals exhibit none or just one incompatibility. As herein shown, in paternity testing with 2nd degree relatives such situations can likely lead to LR values capable of complementing the previously obtained with STRs, apparently supporting a (wrong) conclusion of paternity.
For minimising the likelihood of including as the true father his close relative, sets of 150 and 100 BAMs for, respectively, duos and trios, should be sufficient to reduce the probability of finding none or one incompatibility to less than 1/1000, even when 2nd degree relatives are (unknowingly) analysed in a paternity test.
Indeed, our conclusion on the available strategy to face “problematic” results is summarised in two points: (a) to extend the typing to other loci (STRs or BAMs, either auto- or heterosomal, which can be very useful in some situations [
]) until reasonable a priori values are reached, and (b) to compute the overall LR including all available evidence clearly stating the extended theoretical and statistical framework involved.
Finally it should be noted that, due to their low mutation rates, 1st order incompatibilities for SNPs and Indels are extremely more effective in discarding false fathers (in trios) than those by 2nd order. Indeed, under the hypothesis of paternity, 2nd order incompatibilities (the only possible in duos) are always explainable by the much more likely presence of a silent allele which leads to a much lower evidence against paternity than when only a (extremely rare) mutation can explain the incompatibility.
Thus, on the issue of choosing the most powerful BAMs for kinship evaluation, some considerations should be taken into account – see mathematical approach in supplementary material (Section 3). It can be easily proved that, considering incompatibilities globally (i.e. 2nd order incompatibilities in duos and clustering the ones of 1st and 2nd orders in trios), markers with equally frequent alleles are those that maximise the probability of finding such an incompatible configuration of types in the population (both for duos and trios, and alleged father and child relatives or not). Nevertheless, the probability of finding 1st order incompatibilities in trios (for relatives or not) reaches a maximum not for markers with equally frequent alleles in the population but for those with unbalanced allelic frequencies: ≈0.21/0.79 [
This work was partially supported by the Portuguese Foundation for Science and Technology (FCT) through grants SFRH/BD/37261/2007 (NP), SFRH/BD/65633/2009 (ECS) and SFRH/BPD/81986/2011 (RP). IPATIMUP is an Associate Laboratory of the Portuguese Ministry of Education and Science and is partially supported by FCT. CMUP is partially funded by the European Regional Development Fund through programme COMPETE and by the Portuguese Government through FCT (PEst-C/MAT/UI0144/2011). The authors would like to thank two anonymous reviewers for valuable suggestions on this work.
Appendix A. Supplementary data
The following is supplementary data to this article:
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