Abstract: A Prime number is a natural number larger than one which cannot be expressed as the product of two smaller natural numbers. Prime numbers have been shown to have significant applications in nature as shown by the life cycle of cicadas of the genus Magicicada. Determining the exact distances between prime numbers is an important unsolved problem in mathematics. This study investigated the distances between prime numbers generated from nucleotides in a segment of Bacteriophage T4 genome. The digits of Euler’s number e and Pi π, two mathematical constants were found to be encoded spatially between the Viral DNA helical segments comprised of prime numbers.

INTRODUCTION

A positive integer is said to be prime in case it has exactly two positive divisors (Eynden, 2001). The two divisors are one and itself. The prime numbers <20 are 2, 3, 5, 7, 11, 13, 17 and 19. Prime numbers, once thought to be of interest only to pure mathematicians have found application in applied science, mainly in the protocols of information security and cryptography with the development of Public Key Cryptography whose security lies in the difficulty of factoring large numbers into their prime factors.

In Biological science, the emergence of prime numbers as life cycles of cicada of the genus Magicicada (Hoppensteadt and Keller, 1976; May, 1979; Barnett, 1997; Goles et al., 2001; Campos et al., 2004) has been found to be an evolutionary strategy to escape predators. Advances in DNA computing (Stojanovic and Stefanovic, 2003; Okamoto et al., 2004; Margolin and Stojanovic, 2005) have brought biological molecules into the field of computer science and information technology. Deoxyribonucleic acid and other cellular organelles are now employed as input and output in performing routine calculations and building molecular computing devices.

Determination of the exact distribution of prime numbers is an important unsolved problem in mathematics. The question is closely linked with the Reimann hypothesis (Tao, 2007) which predicts a precise formula for the probability distribution of primes. The distances between 20, 000 prime numbers generated from Bacteriophage T4 DNA was investigated with the objective of modeling their distribution. Euler’s number (e) and Pi (π) are two of the most important mathematical constants and a part of Euler’s identity (Table 1):

Table 1:	Showing frequency distribution of distances between prime numbers In T4 Phage DNA

The value of e and π to ten significant figures, respectively are e = 2.718281828 and π = 3.141592653.

MATERIALS AND METHODS

Bacteriophage T4 genome (Miller et al., 2003) was sourced from GenBank, the institutional genome depository with accession number AF158101. Bacteriophage T4 represents the most understood model for modern genomics and proteomics and its study has revealed many insights and paradigms in molecular biology. The numbers of each nucleotide base per DNA helical segment, comprising of ten bases (Weaver, 2005) were counted and recorded in successive helical segments of T4 phage DNA (complement 1’-99, 180’ in the 3’-5’ direction). The prime numbers present as numbers of nucleotides were 2, 3, 5 and 7 and the distances between them in 20, 000 samples was investigated (Fig. 1). T4 Phage DNA helical segments comprising only of prime numbers consists of the following combinations of nucleotide bases: 55, 37, 235 and 2233. The spatial distances between these helical segments was counted and recorded (Table 2).

Table 2:	The spatial distances of digits of e as encoded between the helical chains of T4 DNA comprised of prime numbers

Table 3:	The spatial distances of digits of e as encoded between the helical chains of T4 DNA comprised of prime numbers

Fig. 1:	Histogram showing the frequency distribution of the distances between prime numbers in T4 Phage DNA

RESULTS AND DISCUSSION

The distances between the prime numbers generated from Bacteriophage T4 nucleotide bases followed a binomial distribution.

The mean and standard deviation of a binomial random variable is expressed as: μ = np and √np (1-p) (Weiss, 2002). The distances between T4 Phage DNA helices comprised of prime numbers encoded 12 values of e and 15 values of π (to ten significant figures) (Table 3).

Prime numbers are regarded as the atomic elements of natural number multiplication (Tao, 2007). Prime number research is an expansive field in mathematical number theory, often churning out interesting results such as the discovery of the 45th known Mersenne prime: 2^{43,112, 609}-1 which is the largest discovered prime number till date. This research work attempts to bridge the fields of mathematical number theory with molecular biology.

CONCLUSION

In this study the occurrence of the mathematical constants e and π encoded within DNA molecules also an interesting experimental result, perhaps shows that mathematics is the foundation of all there is even molecular biology.