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[MATH] -distribution (see Fig. ), the central limit theorem will not average out the deviations at only [MATH] In particular, the distribution remains asymmetric. With increasing threshold, this distribution moves to the right. The reason for this is that, in order to reach an increasing |
[MATH] , the tail events of the constitutive or the variable sum or both must be used, but it is “easier” (that is, more probable) to use the constitutive one because it contains more atypical events. In the language of large deviation theory, this is an example of the general principle that “large deviations are alway... |
Hollander:2000, , Ch. I) . In the language of biology, the constitutive antigens are the problem of foreign-self distinction: due to their high copy numbers and incomplete averaging, fluctuations persist that occasionally induce an immune response even in the absence of foreign antigens. This occurs if a T-cell recepto... |
Let us now turn to the picture with foreign antigen present (Figs. 10 (right), 12 13 and Table ). One salient feature here is that the variable stimulation rate behaves exactly as in the self-only case: closely peaked around a small mean, unchanged when [MATH] is imposed. The picture is thus dominated by the interplay ... |
(Fig. 12 , upper left). In particular, the foreign stimulation rate is closely peaked at [MATH] ; only the constitutive background has moved slightly to the right, exactly as in the self-only case. For [MATH] |
(Fig. 12 , upper right), where, according to Fig. , foreign-self distinction sets in, the foreign stimulation rate becomes prominent: the right branch of the [MATH] -distribution now becomes populated, and the associated stimulation rates are large due to the large copy numbers [MATH] involved. |
Nevertheless, for [MATH] , the foreign stimulation rate is close to [MATH] in a sizable fraction of the cases in which an immune reaction occurs – here, the reaction is brought about by the constitutive background, which moves to the right just as in the self-only case (but less pronounced). Fig. 13 shows that the cons... |
[MATH] is increased further (Fig. 12 , lower left), every T cell beyond the threshold displays high stimuli for the foreign antigen, their distribution shifting even further to the right and concentrating near the maximal stimulation rate given by the maximum of the function [MATH] of Eq. ( ), more precisely, by [MATH]... |
(Fig. 12 , lower right) must then be matched by the by now familiar shift of the constitutive background. (This last panel is, however, less biologically realistic since the probabilities involved are too small to be relevant – after all, with about [MATH] different T-cell types, threshold values that yield activation ... |
A further illustration of the onset of self-nonself distinction is presented in Fig. 14 . Here we consider [EQUATION] i.e., the probability that, in a T-cell that is activated in the presence of foreign antigen, the self component alone would have been sufficient for the activation. From [MATH] onwards, this probabilit... |
Conclusion and outlook We have established here a method of LD sampling that allows the convenient simulation of the rare events relevant to statistical recognition in the immune system. Thus a more thorough investigation of these events could be carried out. |
But this is only a first step, and the goal for future work is to use this or related methods to investigate biologically realistic models. Indeed, the toy model considered here, which relies solely on distinction by copy numbers, does serve the aim to illustrate that distinction against a noisy background is, at all, ... |
Acknowledgements It is our pleasure to thank Michael Baake and Natali Zint for critically reading the manuscript, and Hugo van den Berg and Frank den Hollander for helpful discussions. This work was supported by DFG-FOR 498 (Dutch-German Bilateral Research Group on Mathematics of Random Spatial Models in Physics and Bi... |
# Source: arxiv 0902.1506 # Title: Needles in the Haystack: Identifying Individuals Present in Pooled Genomic Data # Sections: all # Downloaded: 2026-03-02T08:42:19.610627+00:00 |
Needles in the Haystack: Identifying Individuals Present in Pooled Genomic Data Abstract Recent publications have described and applied a novel metric that quantifies the genetic distance of an individual with respect to two population samples, and have suggested that the metric makes it possible to infer the presence ... |
Introduction In the recently published article “Resolving Individuals Contributing Trace Amounts of DNA to Highly Complex Mixtures Using High-Density SNP Genotyping Microarrays” |
, the authors describe a method by which the presence of a individual with a known genotype may be inferred as being part of a mixture of genetic material for which marginal minor allele frequencies (MAFs), but not sample genotypes, are known. |
The method is motivated by the idea that the presence of a specific individual’s genetic material will bias the MAFs of a sample of which they are part in a subtle but systematic manner, such that when considering multiple loci, the bias introduced by a specific individual can be detected even when his DNA comprises on... |
defines a genetic distance statistic to measure the distance of an individual relative to two samples, summarized as follows: Consider an underlying population [MATH] from which two samples |
[MATH] (of size [MATH] ) and [MATH] (of size [MATH] ) are drawn independently and identically distributed (i.i.d.) [in , these are referred to as “reference” and “mixture” respectively]. Consider now an additional sample [MATH] ; we wish to detect whether [MATH] was drawn from [MATH] , versus the null hypothesis that |
[MATH] was drawn from [MATH] independent of [MATH] and [MATH] Given the MAFs [MATH] and [MATH] at locus [MATH] for [MATH] and [MATH] , respectively, and given the MAFs [MATH] for sample [MATH] with [MATH] |
(corresponding to homozygous major, heterozygous, and homozygous minor alleles) at each locus [MATH] defines the relative distance of sample [MATH] |
from [MATH] and [MATH] at [MATH] as: [EQUATION] By assuming only independent loci are chosen and invoking the central limit theorem for the large number of loci genotyped in modern studies, the article |
asserts that the [MATH] -score of [MATH] across all loci will be normally distributed, [EQUATION] where [MATH] denotes the average over all SNPs [MATH] |
[MATH] is the number of SNPs, and Eq. exploits the assumption that an individual who is in neither [MATH] nor [MATH] will be on average equidistant to both under the null hypothesis, i.e., [MATH] The article |
proposes using this approach in a forensics context, in which [MATH] is a mixture of genetic material of unknown composition (e.g., from a crime scene), and [MATH] is suspect’s genotype; by choosing an appropriate reference sample for group |
[MATH] , it is hypothesized that large, positive [MATH] will be obtained for individuals whose genotypes are included in [MATH] , and hence bias [MATH] while individuals whose genotypes are not in [MATH] should have insignificant |
[MATH] since they should intuitively be no more similar to the mixture sample [MATH] than they are to the reference sample [MATH] |
In , the authors applied this test to a multitude of individuals [MATH] each of which are present in the samples constructed by them for [MATH] or [MATH] and report near-zero false negative rates. The article concludes that it is possible to identify the presence of DNA of specific individuals within a series of highly... |
and [MATH] were used, false positive rates—significant [MATH] for individuals neither in [MATH] nor [MATH] —are not assessed in practice; rather, they are simply assumed to follow the nominal false-positive rate [MATH] |
given by quantiles of the putative null distribution in Eq. In this manuscript, we expand on by investigating the method’s robustness to several inherent assumptions: |
1. that [MATH] [MATH] , and [MATH] are all i.i.d. samples of the same population [MATH] and hence the difference of MAFs [MATH] and [MATH] in the two samples is small; |
2. that the loci [MATH] are independent, such that the central limit theorem may be invoked in Eq ; and 3. that an individual [MATH] in neither [MATH] nor [MATH] does not have sufficient genotype identity (e.g., via inheritance) to true positive individual [MATH] that [MATH] for enough [MATH] to bias [MATH] |
To investigate the effect of these assumptions, we begin with a statement of the problem that attempts to address, analytically derive the effect of deviations from the assumptions, and empirically explore the accuracy of the method in practice using real and simulated genotype data. We conclude with a discussion of th... |
The results presented here reveal that membership classification via Eq. is sensitive to the choice of [MATH] and [MATH] ; that even a small amount of LD will alter the distribution of [MATH] for null samples; and that individuals who are related to members of [MATH] or [MATH] are frequently assigned significant [MATH]... |
is believed a priori to be present in exactly one of [MATH] or [MATH] . However, although these findings suggest that Eq. may have limited utility to reliably detect the identity of an individual in [MATH] or [MATH] without prior knowledge, it may be valuable for verifying that an individual is not in either sample, an... |
could perhaps be extended to other genetic-similarity problems (e.g., in ancestry inference). Materials and Methods We explore the performance of the method described in |
both analytically and empirically. For the empirical studies, we attempt to classify real and simulated samples into pools derived from publicly available data sources in order to assess the chances that an individual is mistakenly classified into a group which does not contain his specific genotype. The data used in t... |
2.1 Experimental genotypes and MAFs Real-world genotypes from publicly available data sets were retrieved as follows: 2287 samples with known genotypes were obtained from the Cancer Genomic Markers of Susceptibility (CGEMS) breast cancer study. The samples were sourced as described in |
. Briefly, the samples comprised 1145 breast cancer cases (sample group C+) and a comparable number (1142) of matched controls (group C–) from the participants of the Nurses Health Study. All the participants were American women of European descent. The samples were genotyped against the Illumnina 550K arrays, which as... |
Additionally, 90 genotypes of individuals of European descent (CEPH) and 90 genotypes of individuals of Yoruban descent (YRI) were obtained from the HapMap Project |
. In both cases, the 90 individuals were members of 30 family trios comprising two unrelated parents and their offspring. SNPs in common with those assayed by the CGEMS study and located on chromosomes 1–22 were kept in the analysis (sex chromosomes were excluded since the CGEMS participants were uniformly female); a t... |
Minor allele frequencies for case and control groups were computed from the CGEMS genotypes. Publicly-available minor allele frequencies from the 60 unrelated CEPH individuals were retrieved directly from the HapMap Project |
. The distribution of MAF differences for each group may be seen in Fig. 2.2 Simulated Genotypes I To explore the potential for a sample whose genotype is drawn on [MATH] or [MATH] (without being a member of [MATH] or [MATH] ) to be misclassified, five sets of 320 simulated genotypes were created by drawing a genotype ... |
S.1: For each locus in each sample, genotypes were drawn on the CGEMS control allele frequencies for that locus. S.2: For each locus in each sample, genotypes were drawn on the CGEMS case allele frequencies for that locus. |
S.3: For each locus in each sample, genotypes were drawn on the HapMap CEPH allele frequencies for that locus. S.4: For each sample, 50% of the loci were selected at random to have genotypes drawn on CGEMS case frequencies, and the other 50% had genotypes drawn on CGEMS control frequencies. |
S.5: For each sample, 50% of the loci were selected at random to have genotypes drawn on HapMap CEPH frequencies, 25% of the the of the loci were selected at random to have genotypes drawn on CGEMS case frequencies, and the other 25% had genotypes drawn on CGEMS control frequencies. |
2.3 Simulated Genotypes II To further explore the influence of genetic similarity, two other simulation sets were created. Beginning with the MAFs from CGEMS controls, here denoted by [MATH] , we create the first set as follows: |
1. Draw [MATH] from [MATH] to simulate the MAFs of a sample of 1000 individuals; 2. Draw 1000 genotypes on [MATH] to simulate genotypes of 1000 individuals who will comprise [MATH] |
3. Construct 200 genotypes ( [MATH] s) for which [MATH] percent of SNPs are chosen at random to be identical to a specific [MATH] individual (selected at random for each of the 200 samples), and the other [MATH] fraction of SNPs are drawn on [MATH] |
4. Perform step 3 for values of [MATH] in 0.01 increments from 0 to 1, thus generating 100 pools of 200 samples each who bear [MATH] identity to a true-positive individual, and apply Eqs. to classify them against the [MATH] and [MATH] generated in steps 1 and 2. |
A second set is created as follows, also using the MAFs from CGEMS controls as [MATH] 1. Draw [MATH] [MATH] independently from [MATH] to simulate the MAFs of two samples of 1000 individuals each; |
2. Draw 200 genotypes ( [MATH] s) on [MATH] to simulate 200 individuals from a population with MAFs biased toward [MATH] by [MATH] percent; |
3. Perform step 2 for values of [MATH] in 0.01 increments from 0 to 1, thus generating 100 pools of 200 samples each to be classified against the [MATH] and [MATH] generated in step 1. |
By creating these sets, we ensure that we have samples for which all SNPs are independent in [MATH] and [MATH] , and that [MATH] and [MATH] are samples of the same underlying population; the classification can then be observed as a function of the similarity parameter [MATH] in both cases. |
2.4 Classification of real and simulated genotypes The method as described in and summarized in the Introduction was implemented using R |
. Subsets of the real data (Sect. 2.1 ) and simulated data (Sect. 2.2 ) described above were classified in a total of 17 tests, starting with a total of 481,382 SNPs and excluding those which did not achieve a minor allele frequency [MATH] 0.05 in both |
[MATH] and [MATH] for a given test. A summary of the tests is provided in Table . Additionally, a series of 200 tests using [MATH] [MATH] , and [MATH] as described in Sect. 2.3 were performed. |
Results We begin with an analytical exploration of the assumptions underlying Eq. , followed by the results of the tests as described in Methods. |
3.1 [MATH] and [MATH] under the null hypothesis To address the need for a fully rigorous examination of the problem which tries to address, we here attempt to set up an idealized situation to which the theory and methods in |
apply, and consider the properties of [MATH] and [MATH] (Eqs. ) in that setting versus deviations from that setting. Let us assume an underlying population [MATH] with MAFs [MATH] from which samples [MATH] (of size [MATH] ) and [MATH] (of size [MATH] ) are drawn i.i.d. Consider now an additional sample [MATH] . The nul... |
[EQUATION] where the factors of two are a consequence of each sample possessing two independent alleles per locus. In , it is proposed that [MATH] |
(the [MATH] -score of [MATH] across all SNPs) follows a standard normal distribution (Eqs. ). This proposition rests upon two assumptions: namely, that the mean [MATH] across all SNPs under the null hypothesis is zero, i.e., [MATH] in Eq. and that the SNPs [MATH] are completely independent such that we can write the va... |
[MATH] in the denominator of Eq. Below, we consider sources of deviation from [MATH] under the null hypothesis. 3.1.1 Deviations from [MATH] |
In the large-sample limit, under the null hypothesis, [EQUATION] and hence [EQUATION] Intuition might further suggest that since [MATH] and [MATH] are both drawn from binomial distributions which are symmetric about [MATH] , any sampling deviations resulting from finite [MATH] will fall symmetrically, and hence [MATH] ... |
1. that the MAF differences between samples [MATH] and [MATH] [MATH] are small; 2. that the sample sizes [MATH] and [MATH] are not only large, but comparable. |
Because the number of SNPs [MATH] is quite large, slight deviations away from [MATH] have the power to shift the location of the null distribution of [MATH] considerably, rendering [MATH] incomparable to a standard normal unless the true |
[MATH] is known. Consider that the difference in [MATH] with and without the [MATH] assumption is [EQUATION] and that because [MATH] ranges on [MATH] [MATH] . This means that |
[EQUATION] which can be quite large for even small values of [MATH] since the number of SNPs [MATH] is on the order of [MATH] . It is thus essential that [MATH] be known or controllable. |
Dependence of [MATH] on slight differences in MAFs [MATH] Let us begin by writing the difference between MAFs [MATH] and [MATH] at locus [MATH] as [MATH] |
[EQUATION] We can then write [EQUATION] and thus [EQUATION] where [MATH] is [MATH] under the null hypothesis. We next make a simplifying assumption: since [MATH] are the minor allele frequencies and thus [MATH] , and since [MATH] and [MATH] are estimates of [MATH] , with few exceptions we will have [MATH] and [MATH] (e... |
[EQUATION] and hence Eq. 12 may be written [EQUATION] where [MATH] denotes probability and where we have exploited the fact that because [MATH] [MATH] are independent samples of [MATH] [MATH] is independent of [MATH] , i.e., [MATH] . Observing that |
[EQUATION] Eq. 14 becomes [EQUATION] which is readily verified by simulation. Eq. 17 implies that when [MATH] deviates from zero, either due to systematic differences in [MATH] and [MATH] (i.e., violation of the assumption that both are drawn on the same population [MATH] ) or due to sampling variation, the location of... |
will be shifted by an amount equal to [MATH] relative to that under the assumption that [MATH] . It is important to note that the shift is a weighted average of [MATH] ; ie, it depends not only on the differences in MAFs [MATH] but also on [MATH] , and hence it is not sufficient that |
[MATH] , since small [MATH] will be amplified when [MATH] is small and reduced when [MATH] is large. As a result, predicting the deviation away from |
[MATH] to properly calibrate [MATH] requires knowing not only [MATH] , but [MATH] as well. In practice, [MATH] is easily calculated (examples of the distribution of [MATH] for the CGEMS and HapMap CEPH groups are given in Fig. ). On the other hand, knowing [MATH] requires making assumptions about the population from wh... |
Dependence of [MATH] on sample sizes [MATH] and [MATH] The effect of deviations from the second assumption above is intuitively obvious: if [MATH] |
[MATH] will better approximate the underlying population [MATH] and so will be closer on average to a future sample [MATH] The dependence is derived explicitly in the Appendix. |
We can demonstrate this effect by simulation, as shown in Fig. . Here, we begin by creating [MATH] SNP MAFs [MATH] uniformly distributed on the interval [MATH] . From these |
[MATH] , we simulate the [MATH] with sample size [MATH] as given by Eq. (i.e., a binomial sample) as well as 200 independent samples [MATH] with [MATH] as given by Eq. . By simulating |
[MATH] per Eq. as [MATH] is varied and computing [MATH] for each sample [MATH] per Eq. , we can observe the dependence of [MATH] under the null hypothesis (i.e. |
[MATH] ) on the sample size of [MATH] . A plot of the result is provided in Fig. As seen in the plot and derived explicitly in the Appendix, the dependence in this case varies indirectly with |
[MATH] ; as expected based on the intuition above, smaller [MATH] leads to larger values of [MATH] , indicating that [MATH] is closer to [MATH] (the larger, more representative sample of [MATH] ) than it is to [MATH] Although the difference is small, |
[MATH] – given in Fig. (B) – is quite large, which would lead to a high false-positive rate in practice if the [MATH] assumption were used and [MATH] values compared to the presumed null distribution [MATH] . Thus, we see that as [MATH] decreases, the distribution of [MATH] under the null hypothesis diverges from the s... |
3.1.2 Deviations from [MATH] Invocation of the central limit theorem to compare [MATH] to a standard normal distribution (as given in Eq. ) requires that the variance of the mean of [MATH] be estimable by the mean of the variance, ie, |
[MATH] . This, in turn, requires that the [MATH] are uncorrelated. However, if the various [MATH] are correlated—most notably due to linkage disequilibrium—this is no longer true. Specifically, the variance of the mean for [MATH] variables |
[MATH] with variance [MATH] and average correlation [MATH] amongst the distinct [MATH] is given by [EQUATION] In the case where the average correlation amongst the [MATH] ’s is zero, Eq. 18 yields the result which is found in the denominator of Eq. ; on the other hand, [MATH] generates a |
[MATH] multiplicative increase over the correlationless variance. The large number of SNPs [MATH] results in little room for any correlation between them: consider that Eq. 18 dictates that for a modest number of SNPs [MATH] even a very slight average correlation between all pairs of SNPs [MATH] would result in a tenfo... |
simply from [MATH] [MATH] , and [MATH] . Instead, this issue may be addressed by choosing fewer SNPs and assuming that [MATH] is sufficiently small. |
3.2 Results of Empirical Tests To demonstrate the results derived in Sect. 3.1 above, as well as to explore the performance of the method in realistic situations, we carried out the computations described by Eqs. |
for various [MATH] [MATH] , and [MATH] as described in Table Distributions of [MATH] for each of the 17 tests described in Table are shown in the corresponding figures listed in the table. Bearing in mind the fact that [MATH] yields a nominal [MATH] |
[MATH] -value) of 0.05 and [MATH] yields a nominal [MATH] when compared to a standard normal distribution, the vast majority of samples we tested which were in neither [MATH] |
nor [MATH] were misclassified as being members of one or the other group when using the [MATH] threshold for rejection of the null hypothesis; the misclassification rate was also higher than expected when using a nominal [MATH] threshold. The high false-positive rate in practice is attributable to sensitivity to the as... |
and then discuss the possibility of improving them based on our analytical and empirical findings. 3.2.1 Deviation from putative null distribution |
Choice of [MATH] and [MATH] In Sect. 3.1.1 , we saw that [MATH] will depend on the characteristics of the samples [MATH] and [MATH] The effect is demonstrated in the results shown in Fig. In these plots, [MATH] statistics (Eq. are given for all the CGEMS and S.1–S.5 samples for three choices of [MATH] and [MATH] |
[MATH] = HapMap CEPH, [MATH] = CGEMS case; [MATH] = HapMap CEPH, [MATH] = CGEMS control; [MATH] = CGEMS control, [MATH] = CGEMS case. |
The distribution of minor allele frequencies for each of these three groups (CGEMS cases, controls, and HapMap CEPHs) and the distribution of MAF differences for all three pairs of these groups may be seen in Fig . Notably, even though it may reasonably be expected that the HapMap CEPH sample closely resembles the Cauc... |
[MATH] is an order of magnitude larger when HapMap CEPH is used as one of the the groups, leading to non-zero [MATH] via Eq. 17 Additionally, the sample size of the HapMap group is much smaller than that of CGEMS, thus biasing classification of an unknown sample toward the larger (and hence more representative) CGEMS s... |
As expected, using the HapMap CEPHs for [MATH] fails to separate the CGEMS case and control distributions, such that CGEMS controls and cases all yield high [MATH] (and hence would all be classified as cases) when [MATH] = CGEMS cases; the situation is analogous for [MATH] = CGEMS controls (Fig. , top and center left).... |
for which all samples are in either [MATH] or [MATH] . As anticipated, the accuracy of the classification of cases and controls is dependent on the choice of [MATH] and [MATH] |
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