Forensic genetics is still work in progress. However, it's not new. Thousands of 9/11 victims were identified through their DNA after the WTC catastrophe from their remains.
The very first use of forensic genetics was in 1986 in the UK. Two women, raped and murdered in Leicester in similar fashion 3 years apart, triggered an arrest. The suspect (bizarrely, as it turned out) confessed to the first murder, but not the second. The police approached Sir Alec Jeffreys at Leicester University, the then Professor of Genetics, for help. Based on DNA analysis from the murder scenes, the suspect was acquitted. A second man, Colin Pitchfork, was subsequently found guilty of both murders. He had escaped justice initially by volunteering his friend for a DNA sample he purported to be his own.
So how does DNA fingerprinting work? There are several techniques, but the most useful one makes use of short tandem repeats (STRs) present in the DNA sequence, usually comprising 4 bases such as GATT. The length of a STR is unique to each person. Each STR of course occupies a certain locus on a given chromosome, and as each chromosome is paired, each person possesses two STRs for a given locus, usually of varying length. Part of a sequence is denoted with a point (for example GATT GATT GATT GATT GA would be denoted as 4.2). If there are 4 repeats of GATT on one cheomosome, and 7 repeats on its allele, that is denoted as 4/7.
The USA has a national DNA database called CODIS, which contains around 5 million DNA fingerprints, a number similar to that held in the UK. CODIS is based on 13 different STRs, giving a likelihood that a given sequence will be matched of 1 in 1 trillion. The UK uses 10 STRs, giving a likelihood of match of 1 in a billion.
The frequency of a given allele in the population is given by the Hardy Weinberg equation. For a homozygous allele p, the frequency is given as p^2, while for heterozygous alleles p & q, the frequency is 2pq. To find the probability that all the STRs present in a given person could be replicated in another person, you multiply the probability of finding each STR in a given ethnic group. Therefore, the more STRs you use, the greater the certainty that no two individuals would have the same profile.
Sometimes DNA from victims can be badly degraded because of time elapsed, making STR analysis difficult or incomplete (say 8 STRs possible instead of 13). Under such circumstances, geneticists exploit single nucleotide polymorphisms (SNPs). These are single base differences that occur every 100 to 300 bases within the human genome. Two out of every 3 SNPs involve replacing a C with a T. By using enough SNPs, usually around 70 where C and T are likely to occur equally often, you can make it virtually certain that a profile cannot statistically belong to another human being on the planet. Thus, if all 70 SNPs were to occur independently, the likelihood of a match would be 2^70, or around 10^21. In practice, though, many of these do not assort independently, a phenomenon called linkage dysequilibrium, which still leaves us with a very high likelihood.
A third technique is to use mitochondrial DNA, which is inherited from the mother. This is highly preserved from one generation to the next, and it is thought that 5% of all Caucasians have the same mitochondrial DNA. It therefore lacks the discriminatory power of STRs. It doesn't degrade easily however, and has been used to solve historical mysteries, such as the one used to disprove Anna Anderson's claim to have descended from the Romanov family. The same investigation also showed that Prince Philip, the Duke of Edinburgh, was a descendant of the Russian Romanovs.
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