‘Stochastic’ effects at balanced mixtures: A calibration study

Forensic Science International: Genetics 8 (2014) 113–125

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Forensic Science International: Genetics journal homepage: www.elsevier.com/locate/fsig

‘Stochastic’ effects at balanced mixtures: A calibration study Vince L. Pascali 1, Sara Merigioli * Institute of Legal Medicine, Catholic University of Sacred Heart, School of Medicine, Largo Francesco Vito 1, 00168 Rome, Italy

A R T I C L E I N F O

A B S T R A C T

Article history: Received 1 August 2012 Received in revised form 23 May 2013 Accepted 13 June 2013

DNA mixtures are challenging not only at low template DNA level but also at highly balanced quantitative ratio. In this latter case, interpretation may be complicated by the joint action of combinatorial uncertainty and stochastic effects of the PCR. We explore this particular and so far little noticed aspect of mixture interpretation by first providing a complete quantitative combinatorial analysis of the two-person mixture model (2PM) at highly balanced ratio of contributors, and then by carrying out a calibration study of the 2PM model on good quality experimental mixtures. The calibration tests provided the evidence for the existence of irregular distribution of peak heights, that can misguide the correct genotype assignment at high template ratios too. Repeating the experiment, performing Bayesian analysis to the whole evidence and developing a careful joint prediction of all plausible genotype datasets is highly mandatory in these cases, prior to set evidentiary LRs and use them in court. ß 2013 Elsevier Ireland Ltd. All rights reserved.

Keywords: Mixtures Stochastic effects Peak area analysis Genotype prediction

1. Introduction Mixtures give forensic scientists the best part of their casework and a significant share of controversies. Problems of interpretation are experienced when a mixture is critically unbalanced and its profile is altered by heavy artifacts – the allele dropouts and dropins. These ‘stochastic’ effects [1] can seriously misguide the interpretation of the relevant profile. It is therefore not surprising that considerable efforts have been recently devolved to the task of understanding them [2–7] and modeling their treatment [8]. Dropouts/dropins are believed to be extreme consequences of a more general phenomenon – the irregular amplification of DNA molecules – operated by the PCR. When a series of DNA templates are enzymatically amplified in the same test-tube, some molecules are better copied than others, to such a point that the final quantitative distribution of ‘amplicons’ may significantly differ from the original templates ratio. Kelly et al. [9] have recently studied this phenomenon and they have found that it is easily detected at low-template DNA level but it also acts at a wide range of template concentration. As clearly implied by these authors, in order to account for this phenomenon, genotypes should be assigned in a probabilistic way – a procedure that would in turn have a profound effect on the current approach to mixture interpretation.

* Corresponding author. Tel.: +39 0635507031; fax: +39 0635507033. E-mail addresses: [email protected] (V.L. Pascali), [email protected] (S. Merigioli). 1 Tel.: +39 0635507031; fax: +39 0635507033. 1872-4973/$ – see front matter ß 2013 Elsevier Ireland Ltd. All rights reserved. http://dx.doi.org/10.1016/j.fsigen.2013.06.007

Applying quantitative parameters in the analysis of good quality mixtures is therefore an issue that is worthy to explore in much deeper detail. A document of the DNA commission of the ISFG, published years ago [10], pointed out that there are two different ways to address mixtures: the ‘unrestricted combinatorial approach’ and the ‘restricted’ approach. The former relies on the Boolean (binary, or threshold driven) allele calling, enumerates all genotypes combinations that are logically possible and arrange them in the numerator and the denominator of the evidentiary LR. The latter approach is quantitative: it compares the peak heights, it determines how much each genotype is plausible and then it sets evidentiary LRs. The Commission recommended adopting the restricted approach as long as minor/major components are clearly identified, but not when the mixture ratio (MR) of the contributors goes to equivalence – as in that case the quantitative data is no longer informative. More recently, the quantitative strategy of genotype typing has been strongly endorsed by a fresh document of the DNA Commission [11,12]. This document [11] is mainly concerned with the problem of interpreting dropouts/dropin in a quantitative-Bayesian way and the issue of basic mixtures interpretation is purposely deferred to the rules established in the 2006 document. But it should not escape to notice that – from the viewpoint of mixture analysis – the quantitative approach of the latest document is nothing else than the ‘restricted’ approach mentioned in 2006. This reopens the question as to whether the quantitative criteria apply to mixtures with balanced ratio of contributors, too. In the words of the Commission: ‘‘when the contributors are roughly 50:50, then the restricted [approach] can converge to the unrestricted approach at the four peak loci and approaches it at all other loci [i.e.: those with three and two peaks

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of a two persons-mixture]’’ [10]. This is a too synthetic statement. Its complete consequences should be deducted from a systematic exploration of what happens to the four-, the three- and the two character’s states of an elementary mixture (two-person mixture, 2PM) when both contributors tend to the quantitative equivalence and the peak heights analysis is administered. Since the publication of a seminal article by Evett et al. [13] and of a few others [14– 16], not many papers have dealt with quantitative analysis of good quality mixtures. In particular, what is the effect played by stochastic phenomena in this context has been so far a matter of only anecdotal description. In this paper we will try to fill this gap of information. We will first give a description of the peak graphs network generated by each genotype combination at various MRs close to the quantitative equivalence within the simplest example of mixture. Starting from the results of this basic matrix, we will carry out a calibration study – based on the analysis of some experimental mixtures prepared at our laboratory – aimed at investigating the alterations introduced into the to the graphs’ network by the PCR-based typing process.

(2) the forth allele D is missing because of an overlap (AB + BC; AB + AC; BC + AC). Hence: (i) in the homozygous case, notation (2) becomes: 0:5Q Q - case AA + BC: AR p1 ¼ 0:5Q A ¼ AR p2 ¼ Q B ¼ 1; C

A

0:5Q

- case BB + AC: AR p1 ¼ 0:5Q B ¼ AR p2 ¼ B

0:5Q

- case CC + AB: AR p1 ¼ 0:5Q C ¼ AR p2 ¼ C

QA QC QA QB

¼ 1; ¼ 1:

(ii) When there is an overlap notation (2) becomes: QA QC - case AB + BC: AR p1 ¼ Q Q ¼ AR p2 ¼ Q Q ¼ 1; B

- case AB + AC: AR p1 ¼ Q - case BC + AC: AR p1 ¼

B

C

QB A Q C

QB Q C Q A

¼ AR p2 ¼

A

QC A Q B

¼ 1;

QA Q C Q B

¼ 1:

¼ AR p2 ¼ Q

Finally, when a doublet {A,B} is met, then: (1) both alleles overlap (AB–AB); (2) both individuals are homozygous (for example: AA–BB); (3) one is homozygous and one overlaps (for example: AB–AA; AB–BB).

2. The elementary mixture model and the experimental design Hence, with reference to notation (2): 2.1. The 2PM model at the level of crude DNA extract At the intersection of each genetic locus, 2PMs are a partition of two genotypes singled out of the Mendelian population. As long as nothing is known about the trace’s content, all genotypes that can exist in nature have a chance to fall within it, and the prior probability of each genotype to fall in the trace is equal to the Mendelian genotype frequency. However, once the typing procedure is completed and a small subset of individual alleles is called, the trace ends up to become a finite subset where the number of extant genotypes reduces to a very few: those admissible by the pairwise combinations of called alleles. If allele calling operates by a binary rule (a threshold rule), the population of over-the threshold alleles have a uniform, posterior probability of existence within the trace. Then genotypes, genotype combinations and their relative frequencies are deducted from non-quantitative rules (logical deconvolution [17] or logical genotype prediction [18]). An alternative way to predict genotypes starts by noticing that each contributor confers a definite number of his/her cells (i.e.: genotypes, allelic pairs) into a mixture. If different amounts of cells are contributed by each cellular line and if these amounts are counted, then the probability of each allele to be in the mixture remains uniform within each cell line but it becomes non-uniform between different cell line subsets. For a two person mixture, the quantitative ratio between the two cell lines is: MR ¼

u 1u

(1)

where u is the proportion of cells deriving from the first contributor, (1  u) that of the other and MR a random variable. The ratio (allelic ratio, AR) between the four molecule quantities {QA, QB; QC, QD} orderly contributed by two donors {p1, p2} is conversely one by definition: AR p1 ¼

QA Q ¼ AR p2 ¼ C ¼ 1 QB QD

(2)

if referred to a quadruplet set {A,B,C,D}. When a triplet set {A,B,C} is met, then either: (1) the fourth allele D is missing because one individual is homozygous (AA + BC; BB + AC; CC + AB) or

(i) When both overlap: Q Q - case AB–AB: AR p1 ¼ Q A ¼ AR p2 ¼ Q A ¼ 1: B B (ii) When both are homozygous: 0:5Q

0:5Q

- case AA–BB: AR p1 ¼ 0:5Q A ¼ AR p2 ¼ 0:5Q B ¼ 1: B

A

(iii) When one is homozygous and one overlaps: 0:5ðQ Q Þ Q - case AB–AA: AR p1 ¼ Q ðQ B Q Þ ¼ AR p2 ¼ 0:5ðQ A Q B Þ ¼ 1; A

- case AB–BB: AR p1 ¼ Q

A

B

QA B ðQ B Q A Þ

A

B

0:5ðQ Q Þ

¼ AR p2 ¼ 0:5ðQ B Q A Þ ¼ 1: B

A

Factorizing for the MR and the two ARs will give: MR ¼

u 1u



AR p1 AR p2

(3)

When the DNA is extracted and the cellular subsets of alleles mix one another, genotypes can be still reconstructed by following an ordered pair set scheme (the Zermelo Fraenkel axiom 3; [19]). Finding the MR of a 2PM is straightforward as long as the two ARs amount to one. 2.2. ARs and MRs after the PCR reaction After the PCR-STRs laboratory procedure is administered, calculating the MR of a mixture according to Eq. (2) becomes problematic. In fact: (a) Eq. (2), may no longer be true; Kelly et al. [9] have studied it in the particular case of heterozygous fragments and have found that the ‘heterozygous balance’ (or ‘h’ factor) is in fact a random variable; (b) since the PCR is a competitive ‘branching’ process, the ratio between non-allelic templates too is altered by this biochemical process; the underlying copying model is of controversial nature (lognormal, according to Kelly et al. [9]; a gamma model, according to Cowell [20]); (c) in mixtures, alleles can overlap (the same peak can be contributed by different individuals in different amount) and this complicates the reconstruction of allelic relationship – even in the absence of stochastic effects; (d) discriminating between ‘true’ overlaps and homozygous ‘overlaps’ is a priori impossible; and it is unclear to what extent stochastic effects involve the true homozygous state.

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As a consequence, deducting clear-cut genotype combinations becomes problematic at the quantitative equivalence (1:1) and near the equivalence (2:1 and 1:2). Here the smallest ‘stochastic’ change in two ‘allelic’ peaks heights can lead to incorrectly call a genotype. Moreover, since peak pairs alterations are likely to act independently of each other, one genotype calling may be correct whereas another other may remain dubious. 2.3. Three theoretical (pre-PCR) matrices for ‘balanced’ mixtures The behavior of balanced mixtures can be studied by casting the following model: (A) the elementary two-person mixture (2PM) at quantitative equilibrium of contributors is to be addressed as a multiplelevel (alleles ! genotypes ! pairs of genotypes) combinatorial set to work with; (B) non-overlapping {A,B,C,D}, partially overlapping {A,B,C} and totally overlapping {A,B} subsets of characters must be separately considered; all pairwise combinations than can logically (i.e.: according to non-quantitative set theory rules) derive from each subset should be listed (the list is to be found in Clayton et al, [15]); (C) a subset of discrete mixing ratios (MR = 3:1; 2:1; 1:1; 1:2; 1:3) should be considered. Choosing 2:1 and 1:2 gives the chance to explore the effect of these critical dilutions over the reconstruction of genotypes; (D) three different matrices should be built, by setting genotype combinations as rows and MR values as columns (the matrix corresponding to the triplet {A,B,C} case is in Fig. 2; tetraplex {A,B,C,D} and duplet {A,B} matrices to be found in the supplementary material attached to this paper). All graphs generated by the matrix should be drawn – as if they would come out of the molecule counting within the crude DNA extract – and they should be graphically idealized in order to make eye-comparisons straightforward; (E) cases of unique graphs will be those corresponding to just one pairwise combination of genotypes. Each of these genotype combinations will have P(GjE) = 1 where G is the predicted genotype and E the quantitative evidence; (F) conversely, the coalescence of more than one pairwise combination under the same graph should classify the prediction as uncertain. These genotype combinations will be assigned elementary partition probabilities. All graphs intercepted by the matrices can be easily drawn by hand. Alternatively (in reverse logical order) one can initially set ad hoc peak height values and let an Object-Oriented Bayesian network (OOBN) predict the pairwise combinations. The OOBN technology [21,22] and in general the probabilistic expert systems [23] are particularly useful to address this issue. We built and used an OOBN (Fig. 1) that, starting from peak heights quantities automatically predicts the corresponding genotype combinations and their partition probabilities. In either the graphical or the OOBN version, the three matrices should be regarded as the theoretical (pre-PCR) repertory of peak graphs generated by the balanced elementary mixtures at regular intervals around the equivalence. 2.4. ‘Calibrating’ predictions at balanced mixtures The behavior of balanced mixtures shown in the three matrices (see Fig. 2 and supplementary material) is merely theoretical and it needs to be calibrated for the ‘stochastic’ effects. A suitable calibration test consists in working out real mixtures, generating real peak height measurements and predicting

Fig. 1. The basic structure of the Object-Oriented Bayesian Network (OOBN) built to generate genotype predictions and partition probabilities, according to the peak area data of a given epg. The structure shown in figure is reproduced at each of the fifteen loci. The 15 networks are connected by the frac node. Nodes are the following: frac: the target node where a mixture ratio is calculated across all loci; p1gt, p2gt: the two contributor’s genotypes; jointgt: the genotypes to be predicted; Ainmix, Binmix, Cinmix, Dinmix: the alleles observed in the mixture and their peak areas. This network is a simplified version of the OOBN used elsewhere [24]. Parent nodes containing information on the gene frequencies have been eliminated, to let the predictions be solely dependent on the quantitative ratio between peaks.

genotype combinations from this data. The OOBN shown in Fig. 1 can be used to generate predictions and probabilities. This exercise should tell if theoretical and real-life peak graphs return the same genotype combinations. In the experimental part of this study (see Section 3) we will thus: 1. address uncertain graphs; 2. reproduce them experimentally by choosing two DNA samples whose alleles overlap at most loci and mix them at appropriate MRs: process these samples by the usual STRs technology; reconstruct the corresponding subsets of genotype combinations by using the OOBN; give them a probability; 3. compare predictions in the matrices and experimental predictions, compare both with the true genotype datasets of the donors; 4. report and debrief discordant and/or wrong predictions.

This calibration test will be partial, as only a subset of the matrix’s graphs is covered by the experimental mixtures. However, it will give a realistic idea of what’s the effect of stochastic phenomena on the genotype predictions at balanced mixtures. The opportunity to carry out calibration tests of this kind has been envisioned by Cowell et al. [18]. 3. Materials and methods A blood sample was obtained from two individuals, whose STR profile (Table 1) was known by previous laboratory activity. DNA was extracted and purified according to a standard organic procedure. The content in DNA was determined by averaging three real-time PCR measurements. Then the two mother solutions were diluted down to 30 ng/mL. The final content in DNA was readjusted/reassessed by series of three more real-time PCR measurements. The two solutions were then mixed at big volume (to ensure accuracy of volume sampling) according to the following nominal ratios: 3:1; 2:1; 1:1; 1:2; 1:3. From each final mix, three aliquots were dispensed in as many PCR test tubes, by using a precision pipette. The fifteen samples were used as templates for as many PCR reactions, using the ProfilerPlus AB kit.

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Fig. 2. Peak graphs generated by a 6  5 mixture matrix (the six genotype combinations that lead to a ‘triplet’ state vs. the five MRs of choice). In order to facilitate comparisons between graphs, dosages are portrayed as squared boxes. In four cases, the same graph is generated by different combinations of genotypes (identical graphs have the same color). These graphs are thus uncertain and partition probabilities have to be assigned to the corresponding pairwise combinations of genotypes. The four uncertain graphs, along with the pairwise combinations they admit and the relevant partition probabilities, are reported at the bottom of the figure. The other twenty-one graphs admit just one genotype combination, whose probability is therefore one. (Peak graphs corresponding to the four peak state and the two-peak state to be found in the supplementary material).

Amplifications were carried out in the same experiment by placing the test tubes in the inner wells of the same thermocycler. Separation of the amplicon products was carried out in the same electrophoretic run. Printouts containing the electropherograms (epgs) and the peak area measurements were finally obtained. Peak area calculations and allele calling were left to the sequencer’s resident software (due to the good quality of the templates, no artifactual peaks were registered). To process peak heights values (listed in Appendix A) we designed an ObjectOriented Bayesian Network (OOBN), enabling to calculate ARs/ MRs, to predict the relevant genotype pairs and to give them a probability (Fig. 1). For an overview on the OOBNs and their forensic applications, see [21–23]. In a previous paper, we had used OOBNs to achieve deconvolutions of elementary mixtures [24]. The network used here is a simplified version of our previous network in which gene-frequencies-parent nodes have been eliminated. In

this way, predictions are sensitive to the peak heights statistics only. The fifteen series of EPGs were elaborated individually by cutpasting their peak heights into the OOBN. The experimental part of this work is therefore meant to calibrate the three matrices (see Fig. 2 and the supplementary material) under the conditions operating at the protocol specified above. Other protocols/kits/equipments are likely to give different outcomes. 4. Results and discussion 4.1. Graphs and their corresponding genotype pairs at the elementary mixture The reference matrices (Fig. 2 and supplementary material) is the collection of all admissible graphs at the five MRs of choice, as

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Table 1 The two individuals whose DNA samples were used to build an experimental mixture. Loci are arranged according to the logical scheme of combination they fall in (see alphabetical notations). One doublet, ten triplets and four tetraplets are represented. Triplets are present both in the overlapping (heterozygous + heterozygous) and in the non-overlapping (homozygous + heterozygous) status. Sistema

Contributor 1

D7S820 D8S1179 D21S11 D2S1338 CSF1PO D5S818 D3S1358 D16S539 D13S317 TPOX vWA THO1 D19S433 D18S51 FGA Amelogenin

9 13 31 17 10 12 15 11 8 8 17 6 13 12 21 X

Contributor 2 10 14 31,2 22 11 13 17 11 8 8 17 8 15 15 26 Y

10 13 31 17 9 11 18 10 10 10 16 7 12 13 24 X

Scheme of combination 10 15 32,2 18 11 13 18 13 11 12 19 9 14 17 25 X

they would show up at the elementary mixture if stochastic effects did not exist. In these graphs, allelic dosages are sketched as square boxes to let the resulting peak shapes be easily comparable. The key feature emerging from them is that at three critical MRs (1:1; 2:1; 1:2) there exist graphs admitting just one genotype combination and graphs that account for several alternative combinations. Obliged combinations have probability one. Alternative combinations have partition probabilities. For example: at MR 1:1, the four-allele set {A,B,C,D} generates six genotypes {AB, AC, AD, BC, BD, CD}, three unordered combinations {AB + CD; AC + BD; AD + BC} and six ordered combinations {AB + CD; CD + AB; AC + BD; BD + AC; AD + BC; BC + AD}. One graph only is admissible and it accounts equally well for the three unordered (uniform probability P = 0.33) or the six ordered (uniform P = 0.1666) combinations; at MR 2:1 the same set of alleles {A,B,C,D} generates three alternative graphs, each admitting just one (‘obliged’) genotype combination; for instance, at the graph in which A and B have same height, the subset {AB; CD} coincides with the whole set of alternative genotypes (its probability is one) whereas {CD + AB} = Ø; {AC + BD} = Ø (both are empty sets, with zero probability).

As shown in Fig. 2 (and supplementary material), when one graph corresponds to more than one genotype combination, the combinations have partition probabilities. When there is a one-toone correspondence between graphs and genotype combinations, the probability is one (uncertain combinations have color-filled peak graphs). According to the three matrices, balanced pre-PCR mixtures should be consequently interpreted as follows:  At MR 1:1:  the quadruplet set {A,B,C,D} admits just one graph (sameheight peaks, here tentatively denominated ‘blackbox’) and three genotype combinations with uniform probability;  the triplet set {A,B,C} has three different graphs, each with two alternative genotype combinations;  the doublet set {A,B} generates a blackbox, with equal-level peaks admitting two different genotype combinations; this graph is reiterated at whatever MR value;  specular combinations (same genotype pair but two haplotypes) exist at several levels. This ‘symmetry uncertainty’ is

AB AB AB AC BC BC AB BB AA AA BB AC BD AC AD

BB AC AC AB AC AC CC AC BC BC AC BD AC BD BC

Mixture dataset

MIX scheme

9,10 13,14,15 31,31.2,32.2 17,18,22 9,10,11 11,12,13 15,17,18 10,11,13 8,10,11 8,10,12 16,17,19 6,7,8,9 12,13,14,15 12,13,15,17 21,24,25,26

AB ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABCD ABCD ABCD ABCD

generally corrected when the overall MR is assigned. The OOBN corrects this uncertainty by the FRAC node facility (Fig. 1).  At MR 2:1–1:2:  quadruplets leave no room to combinatorial uncertainty, as they have three possible graphs each corresponding to one distinctive genotype pair. Symmetry uncertainty persists;  triplets generate one uncertain ‘blackbox’ graph (one-dosage homozygous plus two-dosage heterozygous; this graph corresponds to three different genotype combinations with uniform probability) and nine unambiguous graphs each admitting just one genotype combination;  doublets admits six uncertain out of eight possible graphs, each with two admissible genotype combinations. At MR 1:3 and 3:1, combinatorial and symmetry uncertainties should nominally disappear. Additionally, and in principle, uncertain graphs must only exist at exactly MR 1:1, MR 1:2 and MR 2:1, whereas the narrow corridor in between has unique graphs. However, the matrix has a continuous variability, graphs are hardly recognized and – in real-life mixtures – they tend to swing between different blocks of dubious classification on account of small-size ‘stochastic’ effects As a consequence, the doublet matrix becomes ambiguous in either directions (1:1 $ 1:2 and 1:1 $ 2:1) and the triplets matrix in just one direction (1:1 $ 2:1; see Fig. 3). Quadruplets tend instead to behave more simply: the more distant from MR 1:1 the easier to predict. 4.2. The calibration test A more complex picture emerges at real-life mixtures of the calibration test. Tables 2–6 list all genotype predictions generated by the OOBN (Fig. 1) from the epgs produced at our laboratory (five samples, each in triplicate). It should be noticed that the calibration test is incomplete, as it covers only part of the graphs generated by the three matrices. However, since the two original DNA samples had been chosen in such a way as reproduce most of the uncertain graphs, this test ends up to give a clear indication of what happens when the PCR acts on ‘critical’ profiles. The five experimental mixtures had MRs very close to those theoretically envisioned in the matrices. As a consequence, all discrepancies between theoretical peaks (to be read in the matrix) and real peaks (read in the experiments) indicate that ‘stochastic effects’ are into action. Moreover, whether the underlying phenomenon is sporadic or systematic can be assessed by noticing if it is reproduced in the triplicates.

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Tables 2–6 show that, occasionally, prediction procedures may be absolutely correct but fail to give the highest probability to a ‘true genotype pair’ (i.e. those genotypes that have been actually used to build the mixture, to be read in Table 1). This happens at five loci of the experimental mixture (D3S1358, D16S539, vWA, D13S317, TPOX) and has nothing to do with stochastic effects. Conversely, there are cases in which the ‘true genotypes pair’ is not the best-choice prediction owing to the existence of stochastic effects. Within the MR interval span of 1:3 > MR < 3:1 we noticed that thirteen (out of 225 overall predictions, amounting to 5.8%) of these cases involve significant consequences on the genotype calling (Tables 2–6; to facilitate the interpretation, genotype combinations are reported according to a decreasing probability of occurrence; ‘true genotype pairs’ are typed against a yellow background; wrong predictions caused by stochastic effects are in boldface typeset). If a purely Boolean choice were adopted in these cases, the graphs would probably generate a wrong genotype calling – a circumstance that has a close analogy with the act of calling a wrong genotype at a dropins/dropouts-infested low-template mixture. ‘Stochastic effects’ are more widely represented than the phenomenon of incorrect genotype calling can tell. These effects can for example enhance a ‘true genotype’ calling (see for example triplicate 3 at CSF1PO and D8S1179 MR 1:1) and pass unnoticed. And in fact, due to the random nature of the quantitative changes involved, the practical consequences of stochastic effects on mixture interpretation are unpredictable. Although these effects are easily perceived at the raw data level (see Appendix), studying them at the deconvolution level gives a clearer picture of their consequences on the genotype calling. The following circumstances become therefore of general interest: (A) profiles with non overlapping alleles are less vulnerable than partially and totally overlapping profiles. These profiles dilute the same load of stochastic phenomena over a higher number of peaks. As a consequence, quadruplets are easier to interpret than triplets and doublets; (B) however, every allele subset can mislead, when dosage uncertainties combine with stochastic ambiguity (quadruplets at MR 1:1; triplets with a one-dosage homozygous/two dosage heterozygous); (C) under the stochastic variation, triplets and doublets sway from one uncertain graph to another (intrinsic ambiguity of balanced mixtures [24]) accounting for genotype predictions at mediocre probability; (D) calculating the MR at loci with non-overlapping alleles is much safer than at partially overlapped data subsets; mixture with a higher share of non-overlapping datasets are much easier to interpret. 4.3. Stochastic effects: a heterogeneous phenomenon Stochastic phenomena are often thought to be the byproduct of some imperfection of the biochemical pathway that leads to

8 10 12

Fig. 3. The case of a stochastic effect at a mixture locus with ‘confortably unbalanced’ MR (about 5–1). The mixture was experimentally created by blending individual 8–10 (five dosages) and individual 8–12 (one dosage). Peak areas read as follows.

A B C

18623 19261 3941

Brought to the OOBN of Fig. 1, this data generated the following combinations and probabilities: TPOX 10–12 8–10 0.5783 8–12 8–10 0.4148 BC AB AC AB If called under a binary criterion, the genotype combination (10–12 + 8–10) would have been wrong. There is a considerable uncertainty at the deconvolution, recommending extreme precaution when setting an evidentiary LR at this locus.

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Table 2 Calibration of the experimental mixture (Table 1 at MR 1:1). Three aliquots of the master solution were experimentally processed; each gave an epg and a peak heights dataset. Each row of the table corresponds to a combination of genotypes predicted by the OOBN in Fig. 1 (place-ordered combinations are shown: for example: AB + CD and CD + AB are regarded as different states; the place order is essential to correctly reconstruct the two haplotypes according to a coherent dosage). Predictions are shown according to the same logical order as in Table 1. To facilitate cross-reference with the three original matrices (Figs. 2–4), the peak graph conformation corresponding to each prediction is reported in the last column on the right. If excepting the doublet locus, multiple genotype combinations are generated at each locus prediction. The reconstruction of the haplotypes is therefore laborious and several equally probable specular haplotypes are possible. Symmetry uncertainty is extensively present.

Expected genotype predictions (i.e.: predictions to expect if stochastic effects were null) are shown at the top green row of each bloc of experimental predictions. Correct (same as expected) and true (same as the real genotypes, to be read in Table 1) predictions are signaled in a yellow background. Stochastic effects with significant consequences on the allele calling are on boldface typeset. These cases would trigger a wrong calling under a binary genotyping procedure.

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Table 3 Calibration of the experimental mixture (Table 1 at MR 1:2). Arrangement, graphics, signs and symbols are as in Table 2. Uncertainty in the predicted pairwise combinations is noticeably reduced with respect to Table 2.

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Table 4 Calibration of the experimental mixture (Table 1 at MR 2:1). Arrangement, graphics, signs and symbols are as in Table 2. Uncertainty in the predicted pairwise combinations is noticeably reduced with respect to Table 2.

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Table 5 Calibration of the experimental mixture (Table 1 at MR 1:3). Arrangement, graphics, signs and symbols are as in Table 2. Uncertainty in the predicted pairwise combinations is noticeably reduced with respect to Table 2.

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Table 6 Calibration of the experimental mixture (Table 1 at MR 3:1). Arrangement, graphics, signs and symbols are as in Table 2. Uncertainty in the predicted pairwise combinations is noticeably reduced with respect to Table 2.

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amplify a blend of DNA substrates – and supposed to exist only at certain particular analytical situations (for example, the low-copy number profiling). However, in a broader sense, stochastic effects may derive from a series of other phenomena – for example, all those aspects of the laboratory procedure involving measurement errors. Discerning between PCR and non-PCR effects may be therefore difficult and is eventually un-necessary. Additionally, stochastic effects can indifferently affect low-template and high-quality profiles, balanced and unbalanced mixtures, and the relevant procedure of genotype calling may be misled in all of these analytical contexts. An example is in Fig. 3. This figure shows a good quality profile referring to an experimental trace whose two contributors mix at a quite comfortable MR  5:1. The major (8–10) and the minor contributor (8–12) partially overlap at allele 8, but the resulting epg conveys the wrong information that the overlap at allele 10. This profile would be incorrectly interpreted as 8–10 + 10–12. 4.4. Correcting for the stochastic effects? We have provided an evidence that stochastic effects are into action when a well balanced mixture is processed via the usual PCR-fluorescence detection procedure. The relevant problem thus arises of how to deal with a phenomenon that can be a serious obstacle to the reconstruction of the true genotypes at profiles of apparently good technical quality. Coming to terms with a problem like this implies to: (a) understand the true nature of the biological phenomenon; (b) model the procedure of genotype prediction by introducing some correction factor to counteract stochastic effects; (c) go back to an extensive calibration work on experimental mixtures and verify that the correction does really work. Concerning the real nature of the stochastic effects, it is unclear whether they conform to a predictable model. The fact that both a gamma model [20] and a log-normal model [9] have been recently regarded as plausible models calls in itself for further experimental work. Additionally, one should notice that, in a statistic of peak heights, stochastic variability is introduced by more trivial factors. Finding a simple way to correct of it would be therefore of dubious practical value. Deconvolutions are somewhat in themselves a countermeasure. As shown in Tables 2–6, the presence of multiple predictions with mediocre probability of existence warns against the use of a genotype pair that is too uncertain to be incorporated into an evidentiary LR. Predicting all plausible genotype combinations is, in our view, the best way to set the most appropriate evidentiary LR. Passing from predictions to LRs requires elementary computation. The following two cases may be envisioned: (1) obliged combination at a peak area profile. If just one genotype combination is possible, then the probability of a given pair of compatible suspects to be in the mixture is equivalent to the product of their Mendelian genotype probabilities. For example, if a 2:1 MR 2PM with (A,B)2 and (CD)1 asset is compared to a pair of AB and CD suspects, then the probability of a random match is 1/(2pq  2rs); p, q, r, s being the gene frequencies). If CD is assumed to be a priori certain, the probability of suspect AB to be in the mixture is 1/2pq; (2) alternative combinations at a peak are area profile. A LR should be set by placing the compatible asset at the numerator and all possible assets at the denominator. For example: the 2PM asset is (AA)1 + (BC)2 (MR 1:2), two suspects (AA and BC) are available; combinations are: (AA)1 + (BC)2; (BB)1 + (AC)2; (CC)1 + (AB)2 each with 0.33 combinatorial probability. Then: LR = 0.33  (p2 + 2qr)/ [0.33  (p2 + 2qr) + 0.33  (q2 + 2pr) + 0.33  (r2 + 2pq)].

4.5. A remedy to stochastic effects: the joint Bayesian analysis When a genotype pair into a mixture – or a genotype into a simple trace – is uncertain, the analyst may search for a fresh insight into the issue and he/she often collects data from another experiment. This is a suitable way to tackle ‘stochastic’ effects. In fact, as our triplicate experiments show (see Tables 2–6), peak heights variation is nonsystematic and there is much sense in gathering more data on the same evidence. Moreover, unlike their LT DNA counterparts, highly balanced, good quality DNA mixtures give a serious chance to repeat the experiment, sometimes practically at will. The problem thus arises of how to combine the evidence incoming from multiple typing results pertaining to the same trace. Recently, this issue has been much debated and several schemes of evaluation have been proposed, for example: a ‘consensus’ [25], a ‘composite’ [26] and a ‘consensus/composite’ [27] method. Surprisingly, these methods extend a Boolean scheme of analysis to a quantitative problem (i.e.: the genotype calling based on joint evaluation of several peak heights measurements) as if genetic typing were equivalent to tossing a coin or throwing a die. These procedures can be used in the context of a binary allele calling. But if – as it is convincingly vindicated by the ISFG commission [11] – one switches to the Bayesian approach, then Boolean procedures should be dropped out in favor of the joint quantitative Bayesian analysis [24,28,29]. 5. Conclusions Interpreting mixtures implies to cope on one side with the intrinsic combinatorial uncertainty in evaluating some graph conformation, on the other side with some amount of ambiguity introduced by the stochastic effects. We have shown that these latter are more extensively represented in mixtures that previously supposed, and that they have significant effects on the correctness of genotype calling. Both phenomena should be adequately harnessed by replacing the classical binary model of allele calling with a more efficient probabilistic approach [12]. We have here shown that Bayesian deconvolutions are the optimal method to interpret mixtures. Object oriented Bayesian Networks can greatly help to carry out deconvolutions at not only individual profiles but also at multiple datasets [24]. Conflict of interest statement None of the authors (Vince L. Pascali and Sara Merigioli) has a financial or personal relationship with other people or organizations that could inappropriately influence or bias this paper. Acknowledgement The continuing support of our colleagues at the UCSC Forensic Genetics Laboratory (Ilaria Boschi, Francesca Scarnicci, Laura Baldassarri) is thankfully acknowledged.

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