DNA and prime numbers:
DNA works, among others, as a living computer. Let’s look at some
specific aspects in this area, namely the issue related to prime
numbers. Do not forget that even to computers prime numbers play an
important role in control systems, cryptography and computer
development. The nucleotides of adenine (A), thymine (T), guanine (G)
and cytosine (C), which are the building blocks of DNA, having
respective molecular weights of 331, 322, 347, and 307. The numbers 331
, 347, 307 are primes while 322 is complex. With the symbols A, T, G, C
simultaneously we denote the corresponding nucleotides, the respective
bases of the nucleotides and their molecular weights. Based on the above
we find that
A + T = 331 + 322 = 653 (1), which is a prime number.
Α + Τ + G + C = 331 + 322 + 347 + 307 = 1307 (2)
which is also a prime number. Easily we see that
(G + T) - (A + C) = 31 (3)
11G + T - A - C = 31
14 G + 6T - 14A - 6C = 314
141 G + 59T - 141A - 59C = 3141
1415G + 585T - 1415A - 585C = 31415
14159G + 5841T - 14159A - 5841C = 314 159
Each connection gives pi = 3.14159 … with one, two, three, etc.
decimal digits multiplied respectively with 10, 100, 1000 etc. The sum
of the coefficients G and T, respectively, and A and C, are 2, 20,
200, 2000, 20000 etc. A stunning binding of DNA to n.
AT - GC = 53 (4)
A + T + C - G = 613 (5) and
AT - GC + A + T + C = 1013 (6),
which are all prime numbers.
Artificially and by using a computer, one can create DNA chains that
have a sum of molecular weights of nucleotides a prime number. Example:
by the portion of a DNA chain containing scattered overall 29A, 59T, 131
g, 107C, turns out a molecular weight of 106,903 which is a prime
number. The numbers 29, 59, 131, 107 are selected and are prime in this
example and the complementary part of the chain with a total nucleotide
29T, 59A, 131C, 107 G has a molecular weight of 106,213, which is also a
prime number. Of course, what matters for DNA is the sequence of
nucleotides because it determines the
sequence of the protein amino acids based on the genetic code. Only a small percentage of the
genome encodes proteins. Nowadays with the mapping of the human genome
and other organizations, with the help of Biomathematics and
Bioinformatics easily we can make, among others, a research of
information relations with prime numbers, in genes or in larger segments
of DNA. I wonder what is the relation among evolution of genome
information through natural selection and primes?
We finish with something that is very interesting: In summary and
without doing many arithmetic operations, if we remove the link (5) by
the relationship (6) we saw above, resulting
AT - GC + G = 400 ⇒ AT - GC
+ G = 10 (GC) since 347- 307 = 40. Finally turns out the relation
G =
\(\frac{\left(A\cdot T\ +\ 10\ \cdot\ C\right)}{\left(9+C\right)}\) (7)
A relation simple and very important linking the molecular weight
of the nucleotide of guanine with the molecular weights of the other
nucleotides in the DNA. But we do not stop here, because the
relation (7) can often be a primes production ’machine’, like: setting for C and G prime values we find a value for the AT
that can be decomposed into prime factorization. Keeping A a prime
value and T the product of the rest (complex number). Often we find that
the sum A + T + G + C is a prime number, as in DNA. Example: with C =
47, G = 173 we find A = 419 and T = 22. The sum of all of them is 661,
and is a prime number. Another example of a large prime is: C = 311, G =
106213 (from previous example), A = 479 and T = 70950. Their sum 177.
953 is a prime number. Often it is found that the number that turns out
from the relation (4), i.e. AT - GC is prime. The symbols can be defined
by C ’, G’, D ’, T’, if we want to emphasize better differentiation.
The above process has two perspectives. First it is a case in which
there are ”hypothetical DNA molecules” where the molecular weights of
the nucleotides are defined each time with different numbers, the three
(G ’, C’, D ’) being primes and one (T’) complex, where their sum
creates a new prime one, etc. This can be a useful tool in search of
unknown prime numbers, both for mathematicians and amateur researchers
working on groups, with a goal of finding even larger primes. Moreover,
for those who use prime numbers for practical reasons, in control
systems, cryptography, etc. The second view is the numbers derived from
the relation (7), to express nucleotide numbers along a chain or a DNA
segment. Those can be applied to RNA as well.
The RNA is linked to the DNA from which it appears, and with the
proteins produced by the encoded information. Last but not least, the first epigenetic alteration that was detected on DNA, was
the methylation of cytosine. Methylation therein means the addition of
a methyl group, i.e., a small chemical molecule consisting of one
carbon atom and three hydrogen atoms, the total molecular weight of
15. The cytosine acquires a molecular weight of 307 + 15 = 322 and
thymine. This mutation of cytosine which causes various phenomena
(gene inactivation etc.), appears on one hand in the chemical structure (which is the most significant), but also in the equation of the
molecular weight with another chemical basis of DNA. Finally, I hope
this work to provide an opportunity to explore other unknown aspects in the relation of prime numbers and biological phenomena.