Abstract:
Provided is a method for determining the presence or absence of an individual&#39;s DNA in a sample containing DNA from two or more contributors. A panel of a plurality of single nucleotide polymorphisms (SNPs) is used. For each SNP in the panel, it is determined whether the minor allele of the SNP is present in the sample, and whether the minor allele is present in the individual&#39;s DNA. If the number of minor alleles that are present in the individual&#39;s DNA that are also present in the DNA sample is above a predetermined threshold, the individual&#39;s DNA is concluded to be present in the sample. Also provided is an array of DNA molecules for use in the method, as well as a method for estimating the number of individuals contributing to a DNA containing sample.

Description:
This is a National Phase Application filed under 35 U.S.C. §371 as a national stage of PCT/IL2010/001019, filed on Dec. 2, 2010, an application claiming the benefit under 35 U.S.C. §119(e) of U.S. Provisional Application No. 61/266,292, filed on Dec. 3, 2009, and under 35 U.S.C. §119(e) of U.S. Provisional Application No. 61/286,271, filed on Dec. 14, 2009, the content of each of which is hereby incorporated by reference in their entirety. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to analyzing DNA mixtures. 
     BACKGROUND OF THE INVENTION 
     The following prior art publications are considered as being relevant for an understanding of the invention. 
     [1.] N. P. Lovrich, M. J. Gaffney, T. C. Pratt, C. L. Johnson, C. H. Asplen, L. H. Hurst and T. M. Schellberg, National forensic DNA study report, National Institute of Justice (2003) http://www.ncjrs.gov/pdffiles1/nij/grants/203970.pdf. 
     [2.] J. M. Butler, Short tandem repeat typing technologies used in human identity testing. Biotechniques 43 (2007) ii-v. 
     [3.] C. Ladd, H. C. Lee, N. Yang and F. R. Bieber, Interpretation of complex forensic DNA mixtures. Croat. Med. J 42 (2001) 244-246. 
     [4.] G. Peter, C. Neumann, A. Kirkham, T. Clayton, J. Whitaker and J. Lambert, Interpretation of complex DNA profiles using empirical models and a method to measure their robustness. Forensic Sci. Int.: Genetics 2 (2008) 91-103. 
     [5.] A. J. Pakstis, W. C. Speed, J. R. Kidd and K. K. Kidd, Candidate SNPs for a universal individual identification panel. Hum. Genet. 121 (2007) 305-317. 
     [6.] K. K. Kidd, A. J. Pakstis, W. C. Speed, E. L. Grigorenko, S. L. Kajuna, N. J. Karoma, S. Kungulilo, J. J. Kim, R. B. Lu, A. Odunsi, F. Okonofua, J. Parnas, L. O. Schulz, O. V. Zhukova and J. R. Kidd, Developing a SNP panel for forensic identification of individuals. Forensic Sci. Int.: 164 (2006) 20-32. 
     [7.] B. Budowle and A. van Daal, Forensically relevant SNP classes. Biotechniques 44 (2008) 603-608, 610. 
     [8.] N. Horner, S. Szelinger, M. Redman, D. Duggan, W. Tembe, J. Muehling, J. V. Pearson, D. A. Stephan, S. F. Nelson and D. W. Craig, Resolving individuals contributing trace amounts of DNA to highly complex mixtures using high-density SNP genotyping microarrays. PLoS Genet. 4 (2008) e1000167. 
     [9.] J. S. Buckleton, C. M. Triggs and S. J. Walsh, Forensic DNA evidence interpretation, CRC Press, Boca Raton, 2005. 
     [10.] L. A. Foreman and I. W. Evett, Statistical analyses to support forensic interpretation for a new ten-locus str profiling system. Int. J. Legal Med. 114 (2001) 147-155. 
     [11.] The International HapMap Consortium, The international HapMap project. Nature 426 (2003) 789-796. 
     [12.] J. B. Fan, A. Oliphant, R. Shen, B. G. Kermani, F. Garcia, K. L. Gunderson, M. Hansen, F. Steemers, S. L. Butler, P. Deloukas, L. Galver, S. Hunt, C. McBride, M. Bibikova, T. Rubano, J. Chen, E. Wickham, D. Doucet, W. Chang, D. Campbell, B. Zhang, S. Kruglyak, D. Bentley, J. Haas, P. Rigault, L. Zhou, J. Stuelpnagel and M. S. Chee, Highly parallel SNP genotyping. Cold Spring Harb. Symp. Quant. Biol. 68 (2003) 69-78. 
     [13.] Y. Q. Hu and W. K. Fung, Evaluation of DNA mixtures involving two pairs of relatives. Int. J. Legal Med. 119 (2005) 251-259. 
     [14] J. Ragoussis. Genotyping technologies for genetic research. Annu Rev Genomics Hum Genet. 2009; 10:117-33. 
     DNA profiling has become a major tool in the forensic world [1]. The current gold standard for forensic DNA profiling is the sizing of 9-15 short tandem repeat (STR) markers [2]. This method has been found to be very efficient for analyzing DNA profiles from specimens containing DNA from a single individual or a simple mixture of two individuals. However, the identification of an individual in complex mixtures (usually more than two individuals), has proven to be difficult [3, 4]. 
     A number of studies have proposed to use bi-allelic single nucleotide polymorphisms (SNPs) for forensic identification [5-7]. These studies propose using SNPs with allele frequencies close to 0.5 in order to increase statistical power. For a given individual, and it is determined to what extent the individual&#39;s DNA, if present in the mixture can account for any difference in allelic frequencies in the mixture and the population at large. In a recent study, the use of high density SNP microarrays (including 500,000 SNPs or more) was shown to enable the identification of an individual in complex mixtures [8]. That study noted that with the large amount of information on allele frequencies of hundreds of thousands of SNPs, one can identify the presence of a single individual when the genotype of the individual is known for the same hundreds of thousands of SNPs, even if the DNA mixture contains DNA from thousands of individuals. The study was mainly presented in the context of the anonymity of individuals participating in large genome-wide association studies (GWAS). The use of their method for forensic purposes is suboptimal for various reasons. First, the method does not efficiently allow the exclusion of relatives, giving the defense an opportunity to claim that the suspect&#39;s relative rather than the suspect himself is represented in the mixture. Second, the method requires accurate allele frequency data for an appropriate reference population, which in many instances might not be available. Third, genotyping hundreds of thousands of SNPs provides genetic information which might be sensitive with regards to protecting individuals&#39; privacy. Lastly, genotyping hundreds of thousands of SNPs is costly for the routine use in forensic laboratories. 
     SUMMARY OF THE INVENTION 
     In its first aspect, the present invention provides a method for determining whether DNA of an individual, referred to herein as “the suspect”, is present in a sample containing DNA from two or more contributors. In accordance with this aspect of the invention, a panel of SNPs is used, and the presence or absence of the minor allele of each SNP on the panel is determined in the DNA mixture and in the suspect&#39;s DNA. As used herein the term “minor allele” refers to the allele of the SNP having the lowest frequency in a predetermined population among the two or more alleles of the SNP. If the number of rare alleles that are present in the suspect&#39;s DNA that are also present in the DNA mixture is above a predetermined threshold, then the suspect is implicated as a contributor to the mixture. Otherwise, the suspect is not implicated as being a contributor to the mixture. Since genotyping technologies are not error free, it is not necessary to detect all of the suspect&#39;s rare alleles from the panel in the mixture. Thus, as shown below, even if a small number of the suspect&#39;s rare alleles are absent from the mixture, it would still be possible to conclude with high degree of certainty that the suspect&#39;s DNA is present in the DNA mixture. 
     As used herein the tern “random man not excluded” (P(RMNE)) [9] refers to the probability that the DNA of a randomly selected individual which is known to be not present in a DNA mixture is erroneously determined to be present in the mixture (a “Type I error”). A P(RMNE)&lt;10 −9  has been proposed to be an acceptable level of a type I error using a ten-locus STR profile [10]. 
     The inventors have found, for example, that if the MAF of each allele on the panel is in the range of 0.05-0.1 in the predetermined population, then with a panel of 1000 SNPs, any randomly selected individual from the same population will typically carry 100-200 alleles on the panel. For such a randomly selected individual, the probability that all of these 100-200 alleles are present in the DNA mixture is usually below 10 −9  i.e., under these conditions, the P(RMNE)&lt;10 −9 . The inventors have further found that under these conditions, P(RMNE)&lt;10 −9  even when taking into account typical genotyping error rates, which are usually in the range of 0-1% 
     The invention can be carried out using any method of SNP genotyping. Methods of SNP genotyping are disclosed, for example, in [14], and include technologies provided by Affimetrix (“GeneChip”) and Illumina (“Beadchips”). 
     When it can be inferred that all of the contributors to the DNA mixture and the suspect are all from a common subpopulation (such as where the DNA sample was obtained at a remote village, tribe or isolated, reservation, where the suspect resides) a panel may be prepared using SNPs having MAFs in a predetermined range in that subpopulation. In other cases, for example in large cosmopolitan cities, the contributors to the mixture and the suspect may be of various ethnicities. In this case, a panel of SNPs could be used having MAFs in a predetermined range in each of two or more subpopulations. 
     A specific example of SNPs with an MAF of 0.05-0.1% in several racial groups (i.e., Caucasians, Africans and Asians) can be found for example in the table shown in  FIG. 9 , which can serve as a source from which the SNPs of the panel are chosen. 
     SNPs that are as separated from each other in the genome may be used in order to minimize linkage disequilibrium among them. For example, SNPs may be used that are at least 100 Kbp from each other. 
     Thus, in its first aspect, the invention provides a method for determining the presence or absence of an individual&#39;s DNA in a sample containing DNA, comprising:
         a. obtaining a sample containing DNA from two or more contributors;   b. in a panel of a plurality of single nucleotide polymorphisms (SNPs), each SNP having a minor allele in a predetermined population, determining for each SNP whether the minor allele of the SNP is present in the sample;   c. for each SNP in the panel, determining whether the minor allele of the SNP is present in the individual&#39;s DNA;   d. If the number of minor alleles that are present in the individual&#39;s DNA that are also present in the DNA sample is above a predetermined threshold, concluding that the individual&#39;s DNA is present in the sample.       

     In the method of the invention, the number of SNPs in the panel can be selected to satisfy one or more requirements. For example, the number of SNPs in the panel can be selected so that an expected fraction of minor alleles that the DNA of a randomly selected individual from the predetermined population has in common with the panel out of all of the alleles on the panel is in a predetermined range, for example, in a range from 5% to 25%, or from 10% to 20% of the SNPs in the panel. The number of SNPs in the panel can be selected so that the probability that the DNA of a randomly selected individual from the population would not be excluded from the mixture, P(RMNE), is less than a predetermined probability, for example, less than 10 −6 , or 10 −9 . The number of SNPs in the panel can be in the range from 500 to 10,000, or in the range from 1000 to 2000. 
     The SNPs on the panel can be selected to satisfy one or more requirements. For example, the SNPs on the panel can be selected so that the minor allele of each SNP in the panel can have a minor allele frequency (MAF) in the predetermined population in a predetermined range, such as a MAF of each SNP is from 0.01 and 0.2 in the predetermined population. The SNPs on the panel can be selected so that an expected number of minor alleles that the DNA of a randomly selected individual from the predetermined population has in common with the panel out of all of the alleles on the panel is in a predetermined range. This range may be for example from 100 to 200. The SNPs on the panel can be selected so that the SNPs are separated from each other by a predetermined number of base pairs such as 100 Kbp. 
     The SNPs in the panel of SNPs may be selected from the SNPs given in the table shown in  FIG. 9 . 
     The step of determining whether an allele is present in a sample may comprise detecting hybridization between a DNA molecule in the sample and a DNA molecule complementary to the allele. The complementary DNA molecules may be attached to a surface to form an array of DNA molecules. Thus, in another of its aspects, the invention provides an array of DNA molecules for use in the method of the invention. 
     In another of its aspects, the method for estimating the number of individuals contributing to a DNA containing sample, comprising:
         a. for each SNP in a panel of SNPs, determining whether the minor allele is present in the sample;   b. determining a number of individuals contributing DNA to the sample based on the fraction of minor alleles in the panel present in the sample from among all of the alleles in the panel;   c. estimating the number of individuals contributing DNA to the sample based on the fraction of alleles in the panel present in the sample.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order to understand the invention and to see how it may be carried out in practice, embodiments will now be described, by way of non-limiting example only, with reference to the accompanying drawings, in which: 
         FIG. 1  shows the expected P(RMNE) (on a −log scale) as a function of the number of SNPs in the panel used (A); minor allele frequency, MAF, (B); and the number of contributors to the mixture, (C) Within each figure four lines are presented for different combinations of parameters; 
         FIG. 2  shows P(RMNE) as a function of the number of contributors to the mixture, where the red columns represent the expected value with a theoretical panel of 1000 SNPs each with a MAF of 0.075, and the blue columns present the median results of 100,000 simulations with a panel of 1000 different SNPs (with MAFs between 0.05 and 0.1), and the error bars represent the 99% confidence interval as obtained from the simulations; 
         FIG. 3  shows empirical distribution of the number of minor alleles present in a mixture for mixtures comprising 1 to 10 contributors, obtained through 1,000,000 simulations (Y axis represents the probability density values); 
         FIG. 4  shows the median P(RMNE) as a function of the number of contributors to the mixture in which values without relatives in the mixture are presented in the red columns; with one brother in the mixture presented in the blue columns; and with two brothers in the mixture presented in the green columns, calculated from 20,000 simulations using a theoretical panel of 1000 SNPs each with 0.075 MAF and the error bars represent a 99% confidence interval; 
         FIG. 5  shows the median RMNE probabilities as a function of the ratio of CEU and non-CEU contributors in a mixture of eight contributors in which mixtures of CEU and CHB are presented in A, and mixtures of CEU and YRI are presented in B, with values for CEU suspects presented in the blue columns and for non-CEU suspects in the red columns by calculating P(RMNE) using CEU and non-CEU population allele frequencies respectively; 
         FIG. 6  shows expected P(RMNE) as a function of genotyping error rate assuming a power of 99% of detecting an individual present in the mixture, with values for mixtures with 2, 5 and 10 contributors presented in the blue, red and green columns, respectively; 
         FIG. 7  shows the median P(RMNE)a −log scale) as a function of the number of SNPs in the panel used. The upper blue line represents a simple scenario with no complications, whereas all other lines are combinations of 2 or 3 complications; 
         FIG. 8  shows the distribution of allele frequencies of a 1000 SNP panel in CEU, CHB, and YRI populations; and 
         FIG. 9  shows a table of SNPs having an MAF of 0.05-0.1% in several racial groups. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Calculating P(RMNE) 
     Assuming that all SNPs are accurately assessed for the individual (no genotyping errors) and that the presence or absence of each allele is accurately determined in the DNA mixture. Under these conditions, it can be concluded that the individual&#39;s DNA is present in the DNA mixtures if and only if all of the individual&#39;s alleles are present in the DNA mixture. Under this assumption, the probability of excluding an individual whose DNA is actually present in the DNA mixture is zero. 
     When looking at a specific SNPi with possible alleles A and B, the DNA of a randomly selected individual can be determined not to be present in the mixture in two cases:
         (i) The individual&#39;s genotype is BB or AB, and no B alleles are present in the mixture. In this case, the probability of a randomly selected individual being excluded from a mixture comprising n contributors due to SNPi (PEi) equals:
 
 PEi =( p ( Ai ) 2 ) n ×(1 −p ( Ai ) 2 )  (1)
       

     Where p(Ai) is the frequency of the major allele A (the allele with the higher population frequency among the two SNP alleles) at the i-th SNP.
         (ii) The individual&#39;s genotype is AA or AB, and no A alleles are present in the mixture. the PEi in this case is:
 
 PEi =( p ( Bi ) 2 ) n ×(1 −p ( Bi   2 )  (2)
       

     The overall probability of exclusion is the sum of (1) and (2). However, due to the significant difference in allele frequencies, exclusion due to (2) is typically about three orders of magnitude less likely than exclusion due to (1). The contribution of (2) to the exclusion probability can thus be neglected, and (1) will equal henceforth be used to calculate PEi). 
     Consequently, the probability of the DNA of a randomly selected individual not being excluded at site i, P(RMNEi) can be estimated as:
 
 P ( RMNEi )=1 −PEi   (3)
 
     Probability of not being excluded across m sites will then be: 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       RMNE 
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         i 
                         = 
                         1 
                       
                       m 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         PEi 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     (4) represents the P(RMNE) for a given panel of SNPs and a DNA mixture from m contributors. For a specific, simulated or actual mixture, with S sites capable of exclusion (sites exhibiting only the major allele A), the P(RMNE) is: 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       RMNE 
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         i 
                         = 
                         1 
                       
                       S 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         P 
                         ⁡ 
                         
                           ( 
                           Ai 
                           ) 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     This probability is only indirectly affected by the actual number of contributors to the DNA mixture. The number of contributors to the DNA mixture affects the number of sites exhibiting the major allele only, which is the factor that directly affects the P(RMNE) in (5). Equation 5 can be used to calculate P(RMNE) even when the number of contributors is not known. 
     Optimal MAF 
     A MAF of 0.5 (the highest possible MAF) is not optimal as it is highly likely that in a mixture of DNA from several individuals both alleles will be present in the DNA mixture (for example, from two different individuals) thus reducing the ability of the method to exclude individuals. On the other hand, alleles with very low MAF will rarely be present in any suspect, thus again reducing the power of the method. The optimal MAF is the value of P(Ai) for which P(Ei) is maximal and can be obtained by differentiation of (1) by P(Ai), assuming P(Ai)=P(Aj)=MAF for all i,j. Setting the derivative to 0 and solving for P(Ai), results in: 
     
       
         
           
             
               
                 
                   MAF 
                   = 
                   
                     1 
                     - 
                     
                       
                         n 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     where n is the number of contributors. 
     SNP Information and Selection 
     Allele frequencies for all SNPs were obtained from The International HapMap Project [11] for the following populations: Yoruba in Ibadan, Nigeria (YRI), Han Chinese in Beijing, China (CHB) and Utah residents with ancestry from northern and western Europe (CEU). A panel of 1000 SNPs was selected based on HapMap information for the CEU population. All SNPs in the panel had a MAF in the range of 0.05-0.1. SNPs were selected with maximal distances between them. The resulting panel contained 1000 SNP that are at least 1.7 Mbp apart one from the other. 
     Calculating P(RMNE) with Relatives Present in the DNA Mixture 
     As an extreme example of the situation where a relative&#39;s DNA is present in the mixture, the probability of an individual (absent from the mixture) not being excluded when one or two of his brothers are present in the DNA mixture was calculated. To exclude an individual under this scenario, the individual needs to be excluded both by the unrelated individuals and by his brother(s). Since the loci of the SNPs in the panel are close to one another, considerable linkage is expected between markers. The probability of exclusion is dependent on the number of sites for which the suspect has half or full identity by descent (IBD) with his relatives in the mixture. The general equation for calculating the probability of non exclusion of a random brother from a given mixture is: 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       RMNE 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         
                           IBD 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         = 
                         0 
                       
                       S 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           
                             IBD 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                           = 
                           0 
                         
                         
                           S 
                           - 
                           
                             IBD 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             P 
                             ⁡ 
                             
                               ( 
                               
                                 IBD 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               ) 
                             
                           
                           × 
                           
                             P 
                             ( 
                             
                               IBD 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                               ⁢ 
                               
                                  
                                 
                                   IBD 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 ) 
                               
                               × 
                               
                                 
                                   ∏ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   
                                     IBD 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     P 
                                     ⁡ 
                                     
                                       ( 
                                       Ai 
                                       ) 
                                     
                                   
                                   × 
                                   
                                     
                                       ∏ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       
                                         IBD 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         0 
                                       
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         P 
                                         ⁡ 
                                         
                                           ( 
                                           Ai 
                                           ) 
                                         
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where S is the number of potentially excluding SNPs in the mixture (containing only the major allele A), IBD0 is the number of sites among the number of potentially excluding SNPs in the mixture in which all the relatives of the suspect in the mixture have no IBD alleles in common with the suspect. IBD1 is the number of sites in which all the suspect&#39;s relatives in the mixture jointly have exactly one IBD allele in common with the suspect. The probabilities of a certain number of IBD0 sites and IBD1 sites are dependent since they come from the same limited pool of S sites. Hence the probability of fording a certain number of IBD1 sites and IBD0 sites is P(IBD1)×P(IBD0|IBD1). In IBD0 sites, the P(RMNE) is the same as in mixtures with no relative (i.e., as given by 5). In IBD1 sites, the suspect necessarily has at least one A allele since one of his alleles is common by descent with his relatives in the mixture, and all of his relatives were AA homozygotes. The probability that the second allele doesn&#39;t exclude the suspect is the probability of the major allele in the population P(Ai), and across all IBD1 sites: 
               ∏     i   =   1       IBD   ⁢           ⁢   1       ⁢           ⁢       P   ⁡     (   Ai   )       .           
(7 sums a matrix of every possible number of IBD0 sites and IBD1 sites, with their probability of accruing multiplied by P(RMNE) in both IBD0 and IBD1 sites. The IBD0 and IBD1 distributions were obtained through simulations.
 
     Simulations 
     Simulating individuals—Randomly selected individuals were simulated by assigning genotypes at each SNP by sampling alleles from a binomial distribution with allele frequencies as specified in the HapMap database, assuming a Hardy-Weinberg equilibrium. To simulate brothers, initially two random individuals (as before) were simulated to serve as parents. One random haploid genome was generated from each parent assuming a 1% proportion of recombination for 1 Mb of physical distance (two such haploid genomes constitute an individual). Independently repeating this process twice or three times generates two or three brothers. 
     Simulating distributions P(IBD1) and P(IBD0|IBD1) were calculated empirically using (7) and simulating the underlying distributions. For each specific mixture, the suspect and his relatives in the DNA mixture were simulated 1000 independent times. IBD1 and IBD0|IBD1 were approximately normally distributed with means and variances estimated from the 1000 simulations for each case. Then, the probability of each number of IBD1 and IBD0 sites was derived from a binomial approximation of the normal probability density function using the mean and standard deviation of the normal probability density function. 
     Results 
     The Effect of Number of SNPs, MAF and Number of Individuals in the Mixture 
       FIG. 1A  presents P(RMNE) as a function of the number of SNPs used in the panel for different values of MAF (0.1 and 0.05, assuming all SNPs have the same MAF), and a different number of contributors to the DNA mixture (5 or 10).−log(P(RMNE)) is linear with the number of SNPs (P(RMNE) decreased with increasing number of SNPs), thus the number of SNPs have a dramatic effect in the efficiency of the samples used and even a modest increase in the number of SNPs will significantly decrease P(RMNE). For example, if under certain circumstances, P(RMNE)=10 −5 , then doubling the number of SNPs used in the panel, while keeping all other factors unchanged, will decrease the P(RMNE) to 10 −10 .  FIG. 1B  presents P(RMNE) as a function of MAF of the SNPs (again assuming all SNPs have the same MAF) for different sizes of SNP panels (500 or 1000) and a different number of contributors to the DNA mixture (5 or 10). Consistent with (6), MAF has an optimum depending on the number of contributors to the DNA mixture. As can be seen in  FIG. 1B , for 5 or 10 contributors the optimal MAF is 0.09 or 0.05 respectively. (For the case of 2 contributors to the DNA mixture, the optimal MAF is 0.18, not shown in  FIG. 1B .) Mixtures with more than 10 individuals are not common and are relatively uninformative, whereas mixtures with less than 5 individuals can be highly informative even with suboptimal MAF for the SNPs used (see  FIG. 1B ). Therefore, for practical purposes a MAF range of 0.05-0.1 for all SNPs can be appropriate.  FIG. 1C  presents P(RMNE) as a function of the number of individuals in the mixture for different sizes of the SNP panel (500 or 1000) and different MAF (0.05 or 0.1). As the number of contributors to the DNA mixture increases, −log P(RMNE) decreases rapidly. Nevertheless, even with 10 contributors P(RMNE) remains significant under these conditions. 
     The results presented in  FIG. 1  (A to C) are expected probabilities assuming that all SNPs have the same MAF. Typically, a DNA mixture will have a specific P(RMNE) calculated from (5), which deviates somewhat from the expected P(RMNE) as calculated from (4). Therefore, it is important to examine the variance around the expected P(RMNE). Simulations were carried out to estimate P(RMNE) variance. The simulations were run with a panel of 1000 SNPs, each with a different MAF all within the range of 0.05-0.1. The average MAF in this panel was 0.075. A total of 1,000,000 simulations were run (100,000 simulations for each number of contributors to the mixture from 1 to 10). Individuals in the mixture were simulated as described above and P(RMNE) for each iteration of the simulation was calculated using (5).  FIG. 2  presents the results of the simulations for each number of contributors to the DNA mixture (blue columns). The bars correspond to a 99% confidence interval. These results indicate that although an actual result will deviate from the expected P(RMNE) value assuming all SNPs have a MAF of 0.075 (red columns), the deviation is minor and in most instances any actual result will fall close to the expected value. For mixtures of ten contributors, the median P(RMNE) was 1.09×10 −14  and out of 100,000 simulations the worst P(RMNE) obtained was 2.88×10 −11 , well below the required threshold. 
     Estimating the Number of Contributors to a Mixture 
     As stated above, knowing the number of contributors is not required to calculate a specific P(RMNE) in an actual DNA mixture. Nevertheless, in many cases this information is forensically important on its own. The number of contributors will affect the number of rare alleles present so that the number of rare alleles present in a DNA mixture can provide information on the number of contributors to the mixture. The extent that this information can provide accurate information about the number of contributors can be evaluated using simulations. 100,000 simulations were performed for each mixture comprising between 1 to 10 contributors.  FIG. 3  presents the distribution of the number of rare alleles present in the DNA mixture as obtained from the 100,000 simulations for each mixture (with 1 to 10 contributors). The number of rare alleles present in a mixture was normally distributed and the means of these distributions increased with the number of contributors. Thus, the number of rare alleles present in a DNA mixture can be used to estimate the number of contributors to the mixture. With 1-3 contributors, the number of rare alleles present will correctly determine the number of contributors in 99.9% of the cases; with 4-10 contributors the number of rare alleles will correctly determine the number of contributors in 91.17% of the cases, and will estimate the number of contributors within a ±1 range in virtually 100% of the cases. 
     The Effect of Close Relatives in the Mixture 
     Here we look into the effect of the presence of close relatives of the suspect in a DNA mixture. For this, P(RMNE) was calculated assuming that the suspect has one or two brothers present in the DNA mixture. A hypothetical panel of 1000 SNPs was used with a MAF 0.075 for each SNP. The P(RMNE) were calculated using (7).  FIG. 4  presents the effect of the presence of brothers in the DNA mixture on the P(RMNE) of a randomly selected individual with one or two brothers present in the DNA mixture. The median and 99% confidence interval bars were obtained from 20,000 simulations for each mixture type. The median probability that a random brother would not be excluded from a mixture of up to 7 contributors including one brother is less than 10 −9 . For mixtures containing two brothers and up to four contributors to the mixture, the P(RMNE) is less than 10 −6 . It should be noted that a non-exclusion probability of 10 −6  is quite significant when only potential brothers are considered. 
     The Effect of Population Specific Allele Frequencies 
     The analyses conducted in the previous sections assume that the reference allele frequencies for all SNPs are known. This assumption is reasonable when the individuals contributing to the DNA mixture and the suspect are all known to be from a population for which allele frequencies are known and the panel of SNPs was selected accordingly. In forensic work, individuals from distinct sub-populations, emigrants or tourists from different populations may be involved as suspects or contributors to a DNA mixture from a crime scene. Different populations may have different allele frequencies for the SNPs in the panel used, and thus will affect the P(RMNE). As an example, the allele frequency distribution of the panel of 1000 SNPs previously selected from the CEU population, all with a MAF in the range of 0.05-0.1, was examined.  FIG. 8  presents the MAF distribution of these SNPs in YRI, CEU and CHB populations. The MAF distribution in YRI and CHB populations was found to differ significantly from CEU. About 19% were not polymorphic and about 22% had a MAF&gt;0.25. 
     The presence of DNA in a mixture from individuals from different populations can cause two distinct problems. First, its presence in the mixture can increase the number of minor alleles in the mixture thus increasing the P(RMNE). However, this in general does not affect the accuracy of the P(RMNE). Second, if the suspects are arrested based on previous information that they belong to a certain population, then the P(RMNE) regarding those suspects could be calculated based on the allele frequencies in their population. This can improve the accuracy of the P(RMNE), possibly with reduced power. The case where reference allele frequencies for the suspect are unknown can be addressed by assuming “worst case” scenario allele frequencies. 
     In order to study the effect of the presence of DNA from individuals from different populations in a mixture, 130 combinations of mixtures with varying numbers of CEU, CHB and YRI individuals were generated. For each combination 5000 independent simulations were run. The P(RMNE) for each simulation was calculated twice, once with CEU allele frequencies, for a random suspect, and a second time using CHB or YRI allele frequencies, for a suspect from each of those populations respectively.  FIGS. 5A and 5B  present, as an example, the P(RMNE) for mixtures containing 8 contributors with different ratios of contributors. As the number of contributors from a non-CEU population increases, the P(RMNE) of a CEU suspect decreases while the P(RMNE) for a non-CEU increases. This is caused by the fact that many of the SNPs in the non-CEU population have a MAF lower than that in the CEU population reducing the probability that a randomly selected individual from the CEU population is not excluded. A CEU suspect has a median P(RMNE)&lt;10 −9  for any ratio of CEU to CHB or CEU to YRI in any mixture of up to 10 contributors. A CHB or YRI suspect has a median P(RMNE)&lt;10 −9  in mixtures of 8 contributors if the number of CHB or YRI contributors to the mixture is below 4. Another exemplary result from the simulations (not shown) is that, for mixtures of up to 6 contributors, any number of CHB or YRI contributors to the mixture, with any suspect, the median P(RMNE) is &lt;10 −9 . 
     The Effect of Genotyping Errors 
     The analyses required to calculate a given P(RMNE) requires a genotype of the suspect with regard to the panel of SNPs and establishing which alleles are present in the DNA mixture with the same panel of SNPs. Genotyping procedures are not errorless and therefore it is important to consider the effect of genotyping errors. When genotyping an individual (the suspect) standard genotyping errors may occur at known rates for known genotyping platforms. In addition to genotyping errors, a given SNP may not produce a genotype, a factor quantified by the so called “call rate”. For common genotyping platforms, call rates are high and have negligible effects on this process. For example, Illumina&#39;s BEADARRAY™ platform employing the GoldenGate assay has a call rate of 99% and Affymetrix GeneChip® Human Mapping. 10K Array Xba 142 2.0 has a call rate of 92% [13]. The attributed genotyping errors for these technologies are 0.3% and 0.04%, respectively. Note that there is a correlation between call rate and error rate. If one only calls those results that look unambiguous then call rate decreases but so does error rate. Therefore it is safe to assume that the effective error rate for forensic analyses is at the lower end of reported rates, as one can decide to intentionally reduce calling rate and only call the results that seem unambiguous. The reduced calling rate can be compensated by simply increasing the number of SNPs in the panel used. When genotyping the DNA mixture, there are two conceptually different genotyping errors. The first consists of determining the presence of an allele that, in fact, is not present and the second consists of not detecting an allele that is present in the mixture. Both errors are affected by the standard genotyping error. However, the second error is also affected by the effective amount and quality of the DNA in the mixture that carries a given rare allele. If the suspect&#39;s DNA is present in the mixture at a relatively low quantity or quality, then the suspect&#39;s alleles may not be read and the suspect may erroneously be excluded from the mixture. This situation will result in establishing that the suspect is not in the mixture and thus it does not introduce erroneous convictions. Therefore, in this section we will analyze the effect of the standard genotyping error and will keep in mind that obviously one can miss the suspect&#39;s alleles if the quantity or quality of the DNA that he contributed to the mixture is too low. 
     The main problem genotyping errors pose, are wrong exclusions of suspects. This can happen when a rare allele of the suspect is not typed in the mixture, or when a rare allele not present in the mixture is erroneously typed as present in the suspect&#39;s DNA. In order to ensure that real contributors are not always (or too often) excluded, a certain (small) number of “excluding” SNPs should be allowed, while still declaring non-exclusion. An excluding SNP is one that has an allele detected in the suspect and not detected in the mixture. Increasing the number of allowed excluding SNPs results in an increased Type I error (i.e., increased P(RMNE)). Not allowing any excluding SNPs may result in a significantly reduced power for the identification of suspects in a DNA mixture. We assume that the number of wrongly excluding SNPs (denoted X) is binomially distributed with n=the number potentially excluding SNPs (the number of rare alleles in the suspect+the number of SNPs without rare alleles in the mixture) and p=the error rate. We define k as the number of excluding SNPs that should be allowed, and α as the desired power of detection. Then k is obtained by numerically solving the equation P(X≦k)&gt;α. 
       FIG. 6  shows the P(RMNE) for mixtures of 2, 5 and 10 contributors with error rates typical of genotyping platforms, i.e., 0.04% or 0.3%. The appropriate number of allowed excluding SNPs (k) was obtained for α=0.99. This resulted in 4-7 excluding SNPs allowed in the case of a 0.3% error rate and 1-2 “excluding” SNPs allowed in the case of a 0.04% error rate. As can be seen in  FIG. 6 , the effect of genotyping errors (at the error rates used) is quite minor. For all cases studied −log P(RMNE) does not decrease by more than 35% as a consequence of genotyping errors. 
     Combination of Effects 
     In each of the previous sections the effects of each of various factors on the results as investigated independently. In real life situations, a combination of factors can be present. Therefore, the effects of two or more complicating factors analyzed using a simulated DNA mixture with 6 contributors. The following three complicating factors: were used (i) a brother of the suspect is present in the DNA mixture; (ii) A DNA mixture comprising 3 CEU and 3 YRI, with the P(RMNE) calculated for a YRI suspect; and (iii) A genotyping error rate of 0.1% maintaining detection power of 99%. We then calculated the P(RMNE) for all three possible two-factor combinations, as well as for the case where all three factors are present.  FIG. 7  presents the results as a function of the number of SNPs used in the panel. As the scenarios studied become more complex it is harder to reach a desired P(RMNE), say, below 10 −9 . A panel of 1000 SNPs may fail to ensure such a low P(RMNE) in complex scenarios. However, with a panel of 2000 SNPs, the P(RMNE) falls below the desired limit in all of the scenarios here considered.