Introduction In recent decades the advancements achieved in bioengineering have helped us develop a better understanding of the origins from which humans and other living creatures spur. The discovery of the Deoxyribonucleic acid (DNA) is the key to all bioengineering. The DNA is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms and some viruses. The main role of DNA molecules is the long-term storage of information. An allele is one of two or more forms of the DNA sequence of a particular gene. Each gene can have different alleles.

Sometimes different alleles can result in different traits. Occasionally different DNA sequences of alleles will have the same result in the expression of a gene. With the help of mathematics and molecular biology scientists are now able to determine how close genetically different nationalities are compared to one another. The contents of this research paper will demonstrate how Calculus is used to establish the genetic similarities between various populations. The table bellow shows the relative frequencies of four alleles in four different populations, the Eskimo, the Bantu, the English, and the Korean. AlleleEskimoBantuEnglishKorean

A?. 29. 10. 21. 22 A?. 00. 09. 07. 00 B. 03. 12. 06. 21 O. 68. 69. 66. 57 The allele frequency of each of the populations can be expressed as a four dimensional vector. In this exercise we label with “a? ” the vector showing the square roots of the relative frequencies of the alleles in the Eskimo population. Let “? ”,”? ”,”? ” be the corresponding vectors for the Bantu, English, and Korean population. The genetic distance between two populations is defined by the angle between the corresponding vectors. We define the angle between two n-dimensional vectors, ? and ? using the dot product: cos? =(v ?? w ? )/? v ? ?? w ? =(v_1 w_1+v_2 w_2+? +v_n w_n)/? v??? w?? , provided? v?? ,? w??? 0 With the angle formula and the information provided by the allele table we can estimate with a fair amount of accuracy how far apart the four populations are from one another genetically. To prepare we will create a glossary of our vectors and their magnitudes written in the proper mathematical notation. First we list the components of each vector: a ? =v(. 29) i ? +v(. 00) j ? +v(. 03) k ? +v(. 68) h ? b ? =v(. 10) i ? +v(. 09) j ? +v(. 12) k ? +v(. 69) h ? c ? =v(. 21) i ? +v(. 07) j ? +v(. 06) k ? +v(. 66) h ? d ? =v(. 22) i ? +v(. 00) j ? +v(. 21) k ? +v(. 7) h ? Second we find the magnitude of each vector: ?a ? ?=v(? v(. 29)? ^2+? v(. 00)? ^2+? v(. 03)? ^2+? v(. 68)? ^2 )=1 ?b ? ?=v(? v(. 10)? ^2+? v(. 09)? ^2+? v(. 12)? ^2+? v(. 69)? ^2 )=1 ?c ? ?=v(? v(. 21)? ^2+? v(. 07)? ^2+? v(. 06)? ^2+? v(. 66)? ^2 )=1 ?d ? ?=v(? v(. 22)? ^2+? v(. 00)? ^2+? v(. 21)? ^2+? v(. 57)? ^2 )=1 Using the definition, is the English population closer genetically to the Bantus or the Koreans? Explain. First we find the angle between the Bantus and the English population by plugging in the values of their components and magnitudes into the dot product equation for finding an angle between two vectors.

Then we solve for ”?. ” The angle is: ?= cos^(-1)? ((b ? •c ? )/? b ? ?? c ? ? )=cos^(-1)? ((v(. 1) v(. 21)+v(. 09) v(. 07)+v(. 12) v(. 06)+v(. 69) v(. 66))/(1•1))? 10. 272° We use the same process to determine the angle between the Korean and the English population. The angle is: ?= cos^(-1)?? ((d ? •c ? )/(? d ? ?•? c ? ? ))=? cos^(-1)? ((v(. 22) v(. 21)+v(. 00) v(. 07)+v(. 21) v(. 06)+v(. 57) v(. 66))/(1•1))? 19. 857° When we compare the obtained values for both angles we see that 10. 272°