Most SNPs have just two alleles. The most common SNP allele is named the major allele. The other SNP allele(s) are named the minor allele(s). In a mixture profile, the minor allele ratio is calculated as the ratio of minor allele reads divided by the total number of reads. Methods for calculating P(RMNE) have been presented that focus on the mixture SNP loci with no called minor alleles in a mixture profile (e.g., SNPs with minor allele ratios <= 0.001 threshold) ^{2, 3} . The P(RMNE) method described by Isaacson et al. ² was implemented in Sherlock’s Toolkit ⁴ . This formulation enabled P(RMNE) calculations with a small number of dropped alleles for reference profiles compared to mixture profiles. For larger DNA panels, an issue with precision was observed with the Sherlock’s Toolkit implementation, see Figure 1. This method was re-implemented in Java with higher precision libraries in an effort to eliminate the calculation artifacts observed ( Figure 1). The Discrete Fourier Transform-Characteristic Function (DFT-CF) method was implemented with Taylor series approximation of trigonometric functions, named Taylor-32 for 32-bit floating point and Taylor-64 for 64-bit floating point calculations.

The Taylor series library functions were replaced with functions from Mathar’s BigDecimalMath class ( http://www.mpia.de/~mathar/progs/jdocs/org/nevec/rjm/BigDecimalMath.html) to address issues detected with the Taylor-32 and Taylor-64 methods using both 64-bit and 152-bit precision.

Adjacent SNPs may be in linkage disequilibrium such that the alleles of the SNPs have non-random association with each other. The linkage disequilibrium between two SNP alleles is measured by D’ with values between 0 (unlinked) to 1 (fully linked). An adjustment factor for linkage disequilibrium is applied for SNPs ordered by chromosome position. Equation (1) represents the sum of linkage disequilibrium (LD) for the N SNPs with major alleles in a mixture. Adjusting the count of mixture SNPs with major alleles (N) by (E) approximates the number of unlinked SNPs with major alleles in a mixture.

E = ∑ i = 2 N D ′ ( S N P i − 1 : S N P i )

C o m b i n a t i o n ( n , i ) = ( n i ) = n ! i ! ( n − i ) ! = n ( n − 1 ) … ( n − i + 1 ) i ! ( 2 ) P R M N E ( L ) = q 2 ( N − E ) * C o m b i n a t i o n ( N − E , L ) * K L ( 3 ) P R M N E ( 0 ) = q 2 ( N − E ) * C o m b i n a t i o n ( N − E , 0 ) * K 0 = q 2 ( N − E ) ( 4 ) P R M N E ( 1 ) = q 2 ( N − E ) * C o m b i n a t i o n ( N − E , 1 ) * K 1 = q 2 ( N − E ) * ( N − E ) * K ( 5 ) P R M N E ( L ) = q 2 ( N − E ) * C o m b i n a t i o n ( N − E , L ) * K L = q 2 ( N − E ) * n ( n − 1 ) … ( n − L + 1 ) L ! * K L ( 6 ) P R M N E ( L + 1 ) = q 2 ( N − E ) * C o m b i n a t i o n ( N − E , L + 1 ) * K L + 1 = P R M N E ( L ) * ( N − E − L ) L + 1 * K ( 7 )

An alternative to the DFT-CF P(RMNE) method was implemented. A mixture will have N loci with no called minor alleles. Let p be the average minor allele ratio at these mixture loci. Let q be defined as 1 – p such that p + q = 1. SNP panels can be optimized for DNA mixture analysis ^{2, 3} ; the average of the SNP minor allele ratios used for a P(RMNE) calculation can be used to approximate large numbers of individual SNPs with similar minor allele ratios. For an individual with two alleles at a SNP loci the probability for these alleles can be represented as (p+q) ² = p ² + 2pq + q ² = 1. A perfect reference match to a mixture has major:major (MM) alleles at every locus with no called minor alleles in the mixture profile. Mismatches are defined as reference loci with major:minor (mM) or minor:minor (mm) at these mixture loci with no called minor alleles (MM). The number of mismatches is defined as L between a reference and a mixture. Let K be (1 – q ²)/q ² represent the ratio of transition from MM to non-MM (i.e., mM or mm). Let Combination represent the standard statistics combination operation for representing possible SNP loci that mismatch between a reference and a mixture (1). P _RMNE(L) can be estimated by the term for no mismatches, q ^2(N-E), times the possible combinations of L mismatches, Combination(N-E, L), times the transition term K ^L (2) ² . Equation (3) illustrates the calculation for no mismatches (L=0), and (4) for one mismatch (L=1). Consecutive terms can be calculated efficiently for multiple L values as illustrated by (5) and (6). This optimization has the additional benefit of multiplying a large value, (N-E-L)/(L+1), with a small value, K, where calculating (N-E)!/L!(N-E-L)! by itself can stress the precision capability of an implementation for large values for N-E and L. Equation (7) represents the P(RMNE) calculation for 0 to L mismatches.

P R M N E ( 0 t o L ) = ∑ i = 0 L P R M N E ( i )           ( 8 )

Timing for the Sherlock’s Toolkit (Python), Taylor, and Mathar algorithms (Java) were run on an Intel Xeon E5-2609 v2 2.5 GHz dual CPU system with 32 GB RAM. Fast P(RMNE) (Ruby) was run on a MacBook Pro laptop with 2.8 GHz Intel i7, 16 GB 1600 MHz DDR3 RAM, 750 GB SSD hard drive.