Mathematics of Forensic DNA Identification
World Trade Center Project
Extracting Information from Kinships and Limited Profiles
Jonathan Hoyle
Gene Codes Corporation
2/17/03


Introduction
- 2,795 people were killed in the World Trade Center attacks on September 11, 2001.
- 20,000 remains were recovered, the vast majority of which would require DNA matching for identification.
- Existing software tools for DNA identification proved wholly inadequate for the scope and magnitude of this project.


Timeline
- September 17: Armed Forces DNA Identification Lab [AFDIL] asks Gene Codes to update Sequencher(tm) for the Pentagon and Shanksville crashes.
- September 28: Office of the Chief Medical Examiner [OCME] in New York City contacts us for new software.
- October 15: Using the Extreme Programming [XP] methodology, software development is underway.
- December 13: M-FISys (Mass-Fatality Identification System) has its first release to the OCME.
- Since: Weekly releases personally delivered to the OCME, to accommodate rapidly changing requirements.


Identification Technologies
- Technologies used for Identification
-- STR
-- mtDNA
-- SNP
- Methods used:
-- Direct Match to a Personal Effect
-- Kinship Analysis


STR: Short Tandem Repeats
- A repeat of a short sequence of bases (4 or 5)
- For example, at locus position D7S280, it is the four base sequence gata we look for:
 ...gatagatagatagatagatagatagatgtttatctc...
- In the above example, gata is repeated 6 times with a 3-base partial repeat.
- "6.3" is therefore assigned for this allele.
- Being diploid, we have two alleles per locus, thus (up to) two values are stored, e.g. 6.3/8.


STR Frequency
- In 1997, the FBI standardized on 13 STR loci used in the national database, CoDIS.
- Frequency data for each locus/allele value is available for various races.  For example:
Locus:	D16S539	TPOX	D3S1358	FGA	    D7S820	vWA	    D13S317	TH01
Allele:	11/13	8	    15.2	21/13.2	10/11	15.2	11/12	9.3
Freq:	8.55%	39.4%	0.099%	0.796%	14.6%	0.099%	18.2%	9.21%
- Since STR loci are independent, these frequencies can be multiplied: 5.6 x 10^-13
- Likelihood = 1 / Frequency = 1.8 x 10^12


STR Profiles
- M-FISys STR profile contains 16 elements:
-- Amelogenin (Gender)
-- 13 CoDIS Core Loci
-- 2 PowerPlex Loci:
- Penta D
- Penta E
- Minimum Likelihood:
		7.6 x 10^15


STR Likelihood Threshold
- OCME wants a minimum likelihood for identification which ensures a chance of a mismatch to be less than 1 in a million.
- Assuming a population of 5000, what is the smallest n such that a 10n min likelihood yields a mismatch prob < 10^-6 ?
- Since likelihood is the inverse of probability, p = 1 / 10^n
- The probability of no mismatch is q = 1 - p = 1 - 1 / 10^n
- The prob. of no mismatch in 5000 = 1 - q^5000 = 1-(1-1/10^n)^5000
- Thus we have the inequality:
1 - (1 - 1 / 10^n )^5000 < 1 / 1,000,000
- Solving for n we get:
 n > -log10(1 - (1 - 10^-6 )^(1/5000)) = 9.699
- Therefore we set n = 10.


Direct STR Identification
- A victim remain (called a disaster sample) can be identified by direct match if its profile is either:
-- complete and matches Personal Effects (2 modalities)
-- partial with no mismatches, with a likelihood >= 10^10 amongst common loci
- A sample was further investigated if its STR profile likelihood >= 10^10 and with either:
-- a single mismatch only, supported by Kinship
-- mismatches due only to allelic dropout


Partial Profiles
- All STR profiles containing at least 11 CoDIS markers or more will have likelihoods ³ 10^10
- 70% of the victim samples yielded partial profiles (missing at least one CoDIS marker)
- 25% of these partial profiles had likelihood values >= 10^10
- Leaving half of victim samples which cannot be identified through STR means alone (using these parameters).


STR Likelihood: Locus Probability
- Likelihood = 1 / Probability Frequency
- OCME has locus-allele frequency data
- Locus Probability can be first approximated by ignoring population structure and using the Hardy-Weinberg proportions:
p^2  for homozygous alleles:  p = frequency of allele
2pq  for heterozygous alleles:  p,q = frequency of each allele
- Above assumes an infinite population with random mating


STR Likelihood: F
- Because the population is finite, we introduce the inbreeding coefficient F 
- Factoring this into the H-W equations:
p2 + p(1-p)F	for homozygous alleles
2pq(1-F)		for heterozygous alleles
- Because F is very small, 1-F  is close to 1, we round it to remain conservative:
p2 + p(1-p)F	for homozygous alleles
2pq		for heterozygous alleles
- OCME chooses the standard F = 0.03


STR Likelihood: Profiles
- Once we have calculated the probability frequency for each locus, we can calculate the likelihood of the entire profile:
- If Pk (Ak) is the probability of allele A at locus k, we can define the likelihood of STR profile S as:
L(S) = Product(k in Alleles)[1 / Pk (Ak)]
- Note that this works even for partial profiles


STR Likelihood: Race
- OCME has frequency values for four population groups: Asian, Black, Caucasian & Hispanic
- Cannot always rely on reported race, and the race is unknown for a disaster sample
- M-FISys computes the Likelihood value across all four races and chooses the lowest value, just to be on the safer, more conservative side.


STR: Kinship Analysis
- Many times there was not sufficient data to perform an STR direct match.
- Cheek swabs from family members of missing persons are taken, and a pedigree tree in M-FISys can be generated.
- Likelihoods are calculated on victim samples to determine to which pedigree(s) they belong.
- Kinship Analysis was not performed if more than one relative was in the victim list.


Kinship Analysis: Likelihood
- As with direct STR, Kinship Likelihood is:
--  the product of Locus Likelihoods over common loci
--  the Likelihood Ratio >= 10^6
--  calculated across all four races, using the lowest, most conservative value
--  uses frequency data from the OCME
- Analysis was performed for these relations: Parent-Child, Full Sibling, Half Sibling


Kinship Algorithm
- M-FISys uses the Kinship algorithm as implemented by Dr. George Carmody of Carleton University
- Kinship Locus Likelihood defined as:
 k = r2x2 + r1x1 + r0x0
-  where the ri's are relationship proportions:
Parent-Child: r2 =  0   r1 =  1   r0 =  0 
Full Sibling: r2 = 1/4  r1 = 1/2  r0 = 1/4
Half Sibling: r2 = 1/2  r1 = 1/2  r0 =  0 
First Cousin: r2 = 3/4  r1 = 1/4  r0 =  0
- and with p & q the frequencies of the high & low alleles resp., the xi's are defined as:
X2		= p^2	if victim is homozygous and matches an allele
		= 2pq		otherwise
X1		= 0		if relative & victim share no common allele
		= p		if relative homozygous & shares low allele
		= q		if relative homozygous & shares high allele
		= p/2		if relative heterozygous & shares low allele
		= q/2		if relative heterozygous & shares high allele
		= (p+q)/2	if relative & victim are identical
X0		= 1		if relative & victim alleles are identical
		= 0		otherwise


Mitochondrial DNA Analysis
- Some victim samples were so degraded that sufficient STR data was not available for either direct STR match or Kinship analysis.
- mtDNA is hardier material, surviving under conditions which nuclear DNA degrades
- mtDNA is a 16,569-based circular genome.
- It is maternally inherited, and thus not unique.
- 5% of the Caucasian population share the same common mitotype.


mtDNA Analysis
- Mito-typing involves direct sequencing of two highly variable regions of mtDNA.
- The two areas used for mitotyping (HV1 & HV2) are not in a coding region.
- Only a sample's differences from the Anderson Sequence (an internationally accepted standard) need be tracked.
- However, 25% of the WTC victims had no maternally-related kin samples.


Mito Likelihood
- To determine likelihood for a given mitotype, we begin by counting its frequency x in the FBI mtCoDIS data of size n.
- The 95% confidence interval for a population proportion with Binomial distribution is estimated by the formula:
[ m - 1.96s/sqrt(n), m + 1.96s/sqrt(n) ]
 where m is the mean and s is the standard deviation.
- Since the probability p is just the number database hits, we set p = x/n, and so we have m = p and s = Ãp(1-p) .
- Thus we have as the upper bound:  x/n + sqrt(x(n-x))/n .
- If there are no database entries, we use:  1 - a^(1/n)  with a = 0.05
- Likelihood = 1 / Frequency


Introduction to SNP's
- Single Nucleotide Polymorphisms
- Represents single base differences
- Work pioneered by the GeneScreen division of Orchid Biosciences
- By being able to collect data from very short sequences, this technology offers a great deal of hope for the identification of badly degraded samples


SNP Selection
- SNP's occur on average every 100-300 bases within the human genome.
- 2 out of every 3 SNP's involve replacing a C with a T.
- Of these, there is a panel of 70 which are chosen, specifically those in which C and T are equally likely.


SNP Likelihood
- A complete profile of 70 SNP's each with an independent probability of 1/2 would yield a likelihood of match at 2^70 = 10^21.
- The probabilities are independent if the SNP's are unlinked, which we define to be at least 50MB apart.
- Unfortunately, it is not possible to have 70 SNP's 50MB apart in a 3GB genome.


SNP Independence
- A study by Dr. Ranajit Chakraborty of the Center for Genome Information concluded:
-- Allelic dependence is very low: 5.71% as compared to 5% expected by chance alone
-- Average heterozygosity of 46% across three population groups: Causian, Black, Hispanic
-- Despite lack of theoretically independent loci, his study supports the use of this 70 SNP panel for identification purposes


Non-Equiprobable SNP's
- Conservative likelihoods can be calculated even without the assumption of equi-probability.
- All bi-allelic heterozygous alleles have a minimum likelihood of 2, regardless of frequency:
f = 2pq = 2p(1-p) <= 0.5  for all p in [0,1]; Thus L = 1/f >= 2
- The minimum likelihood of a SNP profile containing n heterozygous alleles is thus 2^n.
- As Forensic Mathematician Charles Brenner notes, even if the SNP frequencies were 0.1 and 0.9,  99% of cases will have 10 heterozygous loci out of 100.


M-FISys SNP Form
Combining Technologies for Partial Profiles
- The M-FISys software package is designed for rapid cross-pollination of STR, Kinship, mtDNA and SNP data of DNA samples.
- Consistent or conflicting data in one technology can help determine experimental errors resulting in another technology.
- M-FISys also generates Quality Control reports for finding such inconsistencies.


Combining SNP's & STR's
- By selectively choosing SNP's which are unlinked to each other and existing STR loci, independent likelihoods can be multiplied.
- With the exception of CSF1PO & D5S818, all STR loci are on different chromosomes.
- Thus any unlinked SNP's on an unused chromosome can be included in likelihood calculations.
- STR profiles below threshold are missing >= 3 loci
- Even if only 10 SNP's are used, the likelihood can be increased by 3 orders of magnitude!  (2^10 = 10^+3)


More Information
Gene Codes Forensics
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Ann Arbor, MI  48108
(734) 769-7249
 http://www.genecodes.com


Updated Slides:
 http://www.jonhoyle.com/GeneCodes


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