# The Mathematics Behind the Biochemistry of DNA

The Mathematics Behind the Biochemistry of DNA

Problems concerning the understanding of scientific evidence in forensic science are investigated with reference to measures of improbability related with the presentation of such evidence in an adversarial perspective - **The Mathematics Behind the Biochemistry of DNA** introduction. The investigation includes the use of probabilistic arguments associated with expert scientific testimony in the courts. In Scotland, forensic medicine students were given problems presented in court in order for them to learn about uncertainties. These problems were supported by scientific evidences for (Taroni and Aiken 169).

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How is mathematics a useful tool in understanding such problems? As mentioned above, probability is being used to interpret the data which derived from the scientific analysis, usually done in the laboratory with forensic experts or chemists, with the application of established chemical reactions and powerful methods of chemistry and biotechnology. The theories of probability are proven by the laws of mathematics.

The likelihood of a specific occasion to happen is probability is the chance for an event to happen. It can be written in fractions or as decimals between 0 (an impossible event) and 1 (a sure event). Decimals can be further written as percentages. The probability can be calculated as:

The probability of getting heads in one toss is: p(heads) = 1/(1 + 1) = 1⁄2.

Craven mentioned in his Primer on Probability for Discrete Variables that the Bayes theorem is an effective tool in estimating the likelihood of events. In theories of statistics, some conditions are hard to estimate for its likelihood, but the Bayes theorem solves it. An example of events which are hard to estimate for its probability is the case of the DNA matching. This paper will only give the importance of the Bayes theorem to somehow appreciate the application of mathematics in the filed of statistics.

The DNA fingerprint is the identity of a human being, in which he can be differentiated from other humans. It is commonly encountered on movies concerning the real identity of an individual, to whom must he really belong, who is the real mother of this individual, whatsoever. The last resort to finally end the arguments is the DNA matching. DNA matching has been extensively done since DNA sequencing and matching was discovered.

Let us have a comparison of the evidence and a reference or the sample which is of the known source, such as a hair sample from a suspect. Considering a rape case, we could get seminal fluid from the rape victim or from the crime scene to compare with the DNA present in the blood of the suspect or from the fresh semen obtainable from the suspect if he would allow it.

If the two samples produce an indistinguishable DNA profile, then it is strongly evidenced that the unknown and the known sample came from a common source, and the contributor of the unknown sample is the suspect.

Are we confident of the results? From the Forensic Mathematics of DNA Matching by Brenner, when a DNA profile includes rarely a mixture of traits, we can conclude that the suspect is the contributor. It would be better for us to say that the questionable probably be isolated from the wrongdoing and that he matches only by probability.

Deoxyribonucleic acid or more commonly known as DNA is composed of nucleic acid sequences. The first evidence that DNA is a hereditary material in the body was provided in 1944 by scientist Oswald Avery, Colin MacLoed and Maclyn McCarty. Other features of the DNA like its double helical structure were revealed in 1953 by James D. Watson and Francis H.C. Crick with the help of Rosalind Franklin, an X-ray crystallographer. Watson and Crick had the Nobel Prize in Physiology or Medicine in 1962 with Maurice H.F. Wilkins (). These discoveries led to the extensive research of the applications of DNA in terms of its forensic value.

Instead of knowing the whole sequence of the DNA, a percentage of each allele present in the sample is taken into consideration. This is useful if there are many alleles to consider and there are a lot of suspects to observe. The frequency of each allele as observed in the sample is tallied and divided to the total number of alleles present, not considering what alleles there are. The ratios will be compared to what was seen in the crime scene; then the investigator will have an idea of who the suspect more likely is.

The mathematics behind this is very easy. In this manner we could appreciate chemistry, and especially the simple mathematics underneath. Here presented are simple to complicated mathematical applications which make easy the tasks of biochemists and forensic experts in their fields. Numerical representations will help in concretizing the application of math in DNA matching.

Profile probability of the DNA

A match would occur by chance. The following table will give us the data to support the previous statement. It is easiest to illustrate by example how the probability is determined:

Allele 10 at the locus CSF1PO was found 109 times out of 432 alleles which is equivalent to 216 people. There is enough reason to say that there is a p=0.25 estimated chance that a CSF1PO allele selected at random would be a 10. Furthermore, a 0.31 chance (q) of a random CSP1PO allele to be 11 can be estimated. Before we type the suspect, if we take into account that he is not the source of the evidence, then we may think of him as someone who randomly received a CSF1PO allele from his parents. The chance of receiving either 10 from his mother and 11 from his father is the product of p and q (pq). In this, since the chance for the mother and father can be interchanged since they are just mere chances; 2pq formula arises, which will give genotype frequency locus probability. When the numbers of alleles observed are the same, a formula of p2 is adapted. The locus TPOX illustrates this. The term pp or p2 shows the probability of getting allele 8 from each parent combined. Twenty-eight percent of people have similar TPOX genotype as that of the evidence. There is about 4% chance that a person will have a combined genotype of the two loci. It is derived by multiplying 0.28 by 0.16 or 28% of 16%.

The chance for a multiple-locus genotype is achieved by multiplication – by getting the product of the occurrence of the per-locus genotypes and by multiplying by 2 for the event of a heterozygous locus.

Same calculations applied for THO1 and vWA loci, and taking these alleles into account makes a person chosen in random to have the combined genotype from 4% down to approximately 1/7000 (the actual precise value is 1/7142.86).

The example given distrusts the notion of the DNA tests being the best way to identify a person in the crime scene. This is not to say that the technique is disqualified from all applications but it showed us clearly that this technique has a flaw and is not perfect at all.

The overall profile frequency for this case is 0.00014 or about 1/7000. Therefore, we can say that either the suspect contributed the evidence, or an improbable coincidence happened – the once-in-7000 coincidence that an unrelated person would by chance have the same DNA profile as that obtained from the evidence.

The Prosecutor’s Misleading Notion

Writers in newspapers incorrectly interpret on their own the DNA matching results; in this case, that there is only 1 chance out of 7000 that the semen was left by a person other than the suspect. The prosecutor’s fallacy is being committed this way. It is not true for the reason that it acts as if the DNA evidence alone can be an enough criterion to declare that the suspect might be the donor with great probability, which unreasonably tells us that extra substantiation in the case cannot cause any change.

Logically, we cannot rely solely on DNA evidence alone as verification – we still need additional information, which may be very small as needed. We can assume that the suspect was jailed, or he was an ex-convict, before his DNA analysis, which can be a proof already against him.

The Legal Representative’s Way Out

Brenner also mentioned into his article that the defense sometimes tries to lessen the influence of 7000 to one matching. A nice argument would be that there are many men in the city which could probably have that kind of profile, so there is a small chance that the client is the source of the semen. This is to consider that every man in the city have same chances of being on the place of the crime.

Errors in processing the sample or interpreting the outcome may be a source of forensic DNA testing false positives. A model such as the Bayesian model show how the potential for a false positive affects the factual value of DNA evidence and the satisfactoriness of DNA evidence to meet sufficiently the traditional legal standards for conviction. The Bayesian analysis is contradictory with the “false positive fallacy,” an instinctively appealing but flawed alternative interpretation. The results show the relevance of having accurate data regarding both the random match probability and the false positive probability when looking at the DNA evidence. It is emphasized that disregarding or undervaluing the potential for a false positive can lead to serious errors of interpretation, chiefly when the suspect is identified through a “DNA dragnet” or by searching the database, and that ignoring the true rate of error imposes an important element of improbability about the correctness of the DNA evidence .

Summary

One of the applications of mathematics is the computation of the probability for an event to occur. Probability is used on the other hand to venture on the applications of DNA matching. It is the common notion that DNA matching is a perfect technique in forensic investigations. This paper gives light to the small, yet probable mistake that this technique could bring. This paper will make people appreciate the little application of mathematics in this field, yet really important one. Mathematics as a language of science is being revealed because it is not that emphasized when analysis are made. This will somehow give credit to those biologists and chemists went through in studying the DNA, and especially the mathematicians and statisticians who devoted their lives in this endeavour to widen the horizon for the possibilities of the human mind.

Works Cited

Brenner, Charles H. “Forensic mathematics of DNA matching”. 23 April 2008. <http://dna-view.com/profile.htm>

Craven, Mark. “Primer on Probability for Discrete Variables.” September 2007. 24 April 2008. <http://www.biostat.wisc.edu/bmi576/lectures/intro-probability.pdf>

Franco Taroni, Silvia Bozza and Alex Biedermann. “Probabilistic reasoning in the law. Part 1: Assessment of probabilities and explanation of the value of DNA evidence.” Journal of Forensic Sciences 38(3) (1998): 165-77.