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Applications of Calculus to Forensic Science Emily Cribas Calculus II May 1, 2014
Cribas 2 Abstract Forensic science is an ever-growing field of science that can be further subdivided into: toxicology, anthropology, odontology, mathematics, and many others. Applications of calculus to forensic science can most clearly be seen in the fields of forensic biology and pathology. For calculus in forensic biology, DNA sequences use power series as a powerful tool to compare DNA from crime scenes, for example. To put it in simpler terms, power series are to functions what DNA molecules are to people.1 Specifically for pathologists, calculus is needed to estimate the time of death for victims. Overall, calculus has many applications to many of the subfields and forensics and is often a useful tool in crime scene investigation.
Cribas 3 Introduction Since the beginning of the century and even a little before it, shows such as Law and Order, CSI, NCIS, etc, have made forensic science a subject of public fascination. This newfound popularity of a once unexplored and unknown field has popularized the science by making it seem much more interesting and efficient than it might seem. Just like court shows such as Judge Judy and Judge Mathis, these shows have done these professions an injustice by attracting unknowing college students to the major. In actuality, forensic science is the application of scientific knowledge and methodology to legal problems and criminal investigations.2 It is used to solve crime scene investigations through the use of major sciences such as chemistry, physics, biology, and in this case, mathematics. In most cases, the methods used for this science are time-consuming and thorough, unlike what is portrayed through the media. To enter the forensic science major at Penn State, a student must take introductory chemistry, a chemistry lab, a forensics class covering the essential practices used in the field, and finally, a calculus I course.3 This is the case for many universities with accredited forensic science programs, but why do students in this major need high-level math knowledge to help convict a criminal? In actuality, mathematics has several applications to forensic science. Anthropologists use linear equations and angle measurements to determine things such as the subpublic angle on the pelvis to determine gender.3 Blood spatter analysts use trigonometry and asymptotes to determine angles of incidence and points of origin.4 The forensics fire and explosives unit uses rational equations in gas laws to determine time and type of explosives fired, for example.5
Cribas 4 Basically, mathematics can be applied to any field of forensic science and it is one of the most helpful tools used by scientists in the field.
The Use of Power Series in DNA Sequencing Forensic biologists focus more on DNA sequencing using CODIS, the combined DNA investigation system, to compare DNA sequences already in the database to those discovered at crime scenes for example. DNA, deoxyribonucleic acid, is a hereditary material that is constantly replicated in the nuclei of cells. It is a molecule made up of smaller molecules called nucleotides. These nucleotides each consist of a phosphate group, a sugar group, and a nitrogenous base. The nitrogenous bases can be adenine (A), guanine (G), thymine (T), and cytosine (C). The order of these bases can determine what the DNA sequence of an individual may be. These sequences are used to form genes, which provide instructions to the cell on how to create proteins. Along with this, genes also determine physical characteristics that are inherited from parents. Each of these sequences is different for every single person on the planet. Therefore, it can help find individual characteristics to deduce potential subjects to just one. The entire human genome contains about 3 billion bases and about 20,000 genes.6 Since this can be a haunting and time-consuming task to try and come up with and sequence every single gene by sequencing their respective bases, it is important to find more efficient ways to sequence certain parts of the genome to determine matching DNA. This can be accomplished through the use of calculus, specifically through the use of power series.
“Power series are to functions what DNA molecules are to people.”
Cribas 5 Again, DNA sequences are just nucleotide bases rearranged in different orders. The sequence of nucleotides can be compared to a list of numbers. Both are lists of things, much like a normal sequence (a normal sequence is, in fact, a list of numbers.) Both consist of Taylor coefficients, which are numbers that these sequences consist of. In DNA, however, these numbers can be “bonded” to a basic monomial and added together to form an infinitely long polynomial, the Taylor series, a specific type of power series.7 Each Taylor series, as discussed in class, is unique to functions, and can be used to perfectly approximate a function. By perfectly approximating DNA, the sequence of it can be perfectly known, matched, and even cloned from an individual.8 When learning about Taylor series, it was seen that the series of one function can be easily manipulated and adapted to another function. The Taylor series serves as a “parent” function to the new function, which can be a derivative, antiderivative, or some other similar form of the original function. This perfectly resembles how the similar series of DNA units can be related to offspring, parents, and other family members. The similarities in the “Taylor series of related members can be perfectly calculated as well by calculating both series and subtracting the difference.”9 By determining the Taylor series of a DNA sequence, it can be compared to a standard DNA sequence for the database CODIS. What forensic biologists try to figure out is what the power series of someone’s DNA is, finding the function f(x), for which it is the Taylor series. Here’s a summation of all the comparisons between DNA and Taylor series10:
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The relationship between power series and DNA molecules is clear, but how exactly can technicians use the information to actually sequence part of someone’s DNA? Taylor series are helpful in comparing a more elementary function that can be solved for to any level of accuracy. Once the pattern of the first couple of Taylor coefficients is solved for, a Taylor series can be written for it. Once the Taylor series is written for it, the summations of the series get more and more accurate to what the actual function or original DNA genome actually is. This genome can then be tested to see if it is matching by applying convergence tests to The Use of Differential Equations in Forensic Pathology A different kind of calculus is used my medical examiners and forensic technicians everyday to estimate time of death of a victim. When a victim dies, they experience three stages of death, where the body starts to decompose. The first, livor mortis, occurs twenty minutes to three hours after death. In this stage, the blood settles to the lower part of the body, creating a purplish red discoloration of the skin.11 Algor mortis is the rate at which the body cools after death.12 Determining algor mortis is used to estimate time of death. Notably, if the victim has been dead for more than several hours, the change in body temperature may be so large which could sacrifice the accuracy of the time,
Cribas 7 because after several hours, the body cannot get any colder, or the changes is temperature start to reachequilibrium with the environment. The last stage is rigor mortis, which is when the body starts to stiffen. This process can last about 72 hours.13 After this time, it is futile to try to estimate time of death because the temperature has already reached equilibrium with the environment for an uncertain amount of hours or minutes. Importantly, if the body is discovered only a few hours or less after death, it is easiest, and will yield the most accurate results, to determine time of death by using Newton’s law of cooling:
In this case, k is a constant, u is the temperature of the body discovered at the scene of the crime, and 70 is just an example of what the temperature of the environment could be. As many know, when hot objects are left in a colder atmosphere, the objects, or bodies, in this case, cool down to reach equilibrium with their environment. Newton discovered that this rate of cooling was directly proportional to the difference between the temperature of the object and the environment.14 It can also be inferred from this law that the more of a difference between the object and the body, for example, if the body is extremely hot compared to the environment, the faster the object/body will cool. To solve for Newton’s law of cooling, experience in differential equations and integration techniques is necessary. An example of a problem using this equation is shown below. Basically, the way Newton’s law of cooling works is by calculating the instantaneous rate of change for the temperature of a body. To reiterate, the law states that the rate of change in
Cribas 8 temperature is proportional to the difference between the temperature of the object and that of the surrounding environment. With both temperatures, it is possible to figure out the constant and the rate by means of integration and the use of the properties of exponential functions. The following problem shows how this method works by determining both constants and, notably, by deriving a clear function between the temperature of a murdered body and time to estimate time of death based on initial conditions.15
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Cribas 11 Bibliography 1. http://legal-dictionary.thefreedictionary.com/Forensic+Science 2. http://forensics.psu.edu/resources/faqs 3. http://www.mcs.sdsmt.edu/tkowalsk/portfolio/downloads/pres_UsingTheCSIEffect.pdf 4. http://www.livescience.com/37247-dna.html 5. http://www.math.wpi.edu/Course_Materials/MA1022A96/lab2/node5.html 6. http://www.mcs.sdsmt.edu/tkowalsk/series/Notes-1-Introduction.pdf 7. http://www.mcs.sdsmt.edu/tkowalsk/series/Notes-7-Power-series.pdf 8. http://www.mcs.sdsmt.edu/tkowalsk/portfolio/downloads/pub_FunctionalDNA.pdf 9. http://www.wayzata.k12.mn.us/cms/lib/MN01001540/Centricity/Domain/688/Death%20 Stages.pdf 10. http://w3.uniroma1.it/dsg/enoc2011/proceedings/pdf/Machado_et_al_1.pdf 11. http://www.math.washington.edu/~m125/Source/ws/week9/DiffEQ/sol9.pdf 12. https://www.math.washington.edu/~m125/Worksheets/DiffEQ.pdf
