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\begin{document}
\title{1. Biological phenomena and prime numbers}
\author[1]{Marinos Spiliopoulos}%
\affil[1]{Biology Professor }%
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\date{\today}
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As we know, numbers are involved in the phenomena of life and beyond, in
various ways. Biomathematics and Bioinformatics nowadays are thriving
and will have greater development in the future. Some number in
particular have a lot of interest and give a special charm to both the
inanimate and the living matter. For example, the p and f numbers are
involved in many biological procedures in different ways. Specifically,
p = 3.14159\ldots{}, i.e. the quotient of the circumference of a circle
to its diameter, is revealed in the formulas of the potential difference
in the cell membranes, in the flow of liquid in vessels, etc. The f =
1.618 aka golden ratio appears in various proportions of the human body,
in the way leaves are assembled in plants etc. The Fibonacci sequence is
correlated to f, when calculating rabbit offspring. The sequence and
thus the \selectlanguage{greek}φ, \selectlanguage{english}are connected to the logarithmic spiral that we meet at the
shape of the shells of land or marine organisms such as snails, nautilus
etc.. Something remarkable with the f is the following: If the core
volumes, the cytoplasm and the cell are consecutive geometric
progression conditions it is known that
\par\null
\(\frac{Vkp}{Vp}\) = \selectlanguage{greek}φ \selectlanguage{english}(cell-core constant).
\par\null
Because this may apply for each moment of a cell's lifecycle, given that
volumes can be multiplied by a coefficient (effective volume) we end up
with the relationship
\par\null
~\(\frac{Vkp}{Vp}\)= n [?] \selectlanguage{greek}φ
\selectlanguage{english}\par\null
The disorder of the relationship between the volumes of
nucleus-cytoplasm is of great interested in cases of malignancies,
tumors, etc.~ If somebody looks through a microscope, for instance,
cells and their core, he will notice a huge difference between leukemic
and normal cells. Another case is the number e = 2,711828, the base of
natural logarithms. There is no science field which is not correlated
with this number.
\par\null
In this paper we will deal more thoroughly with the prime numbers in
biology. The prime numbers are integer positive numbers divisible only
by themselves and the unit 1, eg. 2, 3, 5, 7, 11, 13,17\ldots{} The
primes that differ by 2 are called twin primes. Euclid was the first who
proved that the number of primes is infinite. Today we know that the
greater the interval between the numbers is, the more prime sparsity we
have. On the other hand, complex numbers (all positive integers that are
not primes) result as a raw product of primes, e.g. 70 = 2 x 5 x 7 etc.
Therefore any positive integer greater than 1 is either a prime or a
product of primes. Mathematicians for their own reasons, consider the
number 1 neither a prime nor a complex number. Prime numbers, are for
Mathematics what for Chemistry and Physics are atoms of matter: the
building blocks upon which all other numbers are built, the ''atoms''
that compose the Mathematics' universe. Someone said that mathematicians
love prime numbers, such as chemists love atoms and biologists love
genes. The interest in prime numbers is big and despite the era, it
remains at the forefront of scientific activity. Moreover, there are
major unresolved problems over time regarding prime numbers (Riemann
conjecture, Goldbach conjecture, conjecture of the twin primes etc.).
Also, there are no mathematical formulas known to compute the next prime
number, and how many prime numbers are smaller than a specific number.
Two very characteristic sayings of great scientists about the importance
of the above are the following: Hilbert, a great German mathematician
had said that ''if it was possible after 1000 years from my death to
come back to life, the first thing I would ask would be if Riemann's
conjecture has been resolved. '' This said, knowing, as all
mathematicians, that the problem is extremely difficult and the solution
of it will lead to numerous solutions of other mathematical problems.
Paul Erdos, a Hungarian great mathematician who dealt with prime
numbers, just before his death said: ''It will take one million years
until we understand prime numbers.''
Let's have a look at primes' relations with biology, starting with some
simple observations:
\begin{quote}
a) The codons of the genetic code that encode amino acids are 61, while
there are three (dating) that do not encode any amino acid. Each codon
consists of three nucleotides.
b) Among the 37 genes of mitochondrial DNA 13 encode proteins.
c) The smallest form of life, a bacterium that belongs to the mycoplasma
has 521 genes (a prime number).
d) To nucleic acids (DNA, RNA), the orientation of the polynucleotide
chain is 5' - 3'.
e) Microtubules, which are structures of the cellular shell of the
ciliums and flagellums of various eukaryotic cells, consist of 13
parallel protofibrils (with a 5nm diameter each) of alternating
molecules of alpha and beta tubulin.
f) 2 and 3 phosphate groups of nucleotides (ADP, ATP, etc.), 2, 3, 5, N
atoms depending on the nitrogenous base of nucleotides, consist of 5
Carbon atoms therein.
g) 5 radial symmetry in echinoderms, e.g. starfish.
h) 23 chromosomes in human reproductive cell etc.
i) In the genome of prokaryotic organisms, the genes for enzymes
involved in a metabolic way, are organized in operons, i.e. groups that
are subject to common control of their expression. The lactose operon
has three structural genes, many other 2 or 3 structural genes,
tryptophan has 5 etc.
\end{quote}
In order to avoid any misunderstanding, we clarify that in many
biological issues where numerical quantities are involved, there may be
no prime numbers, but can be complex or decimal numbers. However, our
interest in this paper focuses on prime numbers and their correlation
with biological issues. We continue with examples of prime numbers that
appear in natural selection issues and in evolution of organisms. A
known example from the field of the animal kingdom is that two kinds of
cicadas Magicicada tredecim and Magicicada septendecim live in the same
environment with a lifecycle of 13 and 17 years respectively. Throughout
their life, except the last year, they live in soil and feed on the
juices of tree roots. During the last year of their life, for a few
weeks, they transform from larvae to adult, occupy the forest, eat, lay
eggs and die. But how these species evolved so that their cycle of life
is actual prime numbers? An answer is that their mutual emergence to the
forest takes too long and happens once in 13 x 17 = 221 years. While if
their life cycles were complex numbers, eg. 12 and 18 years in the above
time they would have competed six times, as many are the common
multiples of 12 and 18. As we can see, the prime numbers 13 and 17 are
not something abstract and random, but the basis of their survival. Such
examples are likely to exist in the animal and plant kingdom.
The birthday paradox is very interesting. It belongs to the probability
theory and refers to a problem which by the common sense has an unlikely
answer. The formulation of the problem is this: In a group of 23 people
which is the possibility that two among these individuals have their
birthdays on the same day? The obvious answer is 23/365 = 0.063, i.e.
6\%. Despite this, the mathematical solution is 50\%. The possibility is
even 100\% when it refers to 367 people, including those who were born
on February 29. 23 and 367 are prime numbers, and the reference to this
paradox is used because it is related to people (biological entities)
and a biological fact, their birth in a specific time period (birthday).
It is worth mentioning that this paradox is the basis for one of the
most common methods of cryptanalysis, in the corresponding field of
computer science (cryptography).
Prime numbers are related to the molecular basis of apoptosis, or
programmed cell death, which was initially studied in the filamentary
worm Caenorhabditis elegans. Later, it was studied in other
invertebrates and vertebrates. Only in the hermaphrodite filamentary
worm, we have the best of apoptosis study system, to the point that is
considered to be created for this purpose. Apoptosis is a fundamental
biological process that is necessary for the removal of surplus cells
during embryonic development, maintaining the homeostasis in the adult
organism, maturation of cell populations of the hematopoietic and immune
system and malignant transformation.
Typically the apoptosis is not a consequence of disfunction and
therefore differs from necrosis, which is always a result of a harmful
effect. In filamentary worm, during its development when 1090 cells are
produced (hence 1091 total with the original zygote), of which 131
somatic cells are eliminated by the process of programmed cell death
while 3 genes are responsible for it. 1091, 131 and 3 are prime numbers,
and the above constitute an ascertainment, the causes of which are
unknown for now. Maybe, in the future for the entire process of the
phenomenon, emerge also a mathematical model.
\par\null
\emph{Amino acids, proteins and prime numbers:~}The Greek language is
dynamic and mysterious. Perhaps it is the only language in which there
is such a complete identification of signifier and signified. Amino
acids are the building blocks of proteins. The word ``protein'' is
derived from the Greek word ''protos'' (which means ``first''), i.e. the
importance of proteins to the structure and function of cells and
organisms, is of high importance. Let's see how they relate to prime
numbers. Based on their chemical formula 5 amino acids have a molecular
weight which is a prime number. Specifically: Methionine 149, Alanine
89, Leucine 131, Isoleucine 131, Tyrosine 181. The molecular weight of
all the other amino acids is easily calculated and there is no need to
be listed in a table. Because the interesting fact is not the molecular
weights of individual amino acids, but the fact that we can
''artificially'' and very easily design peptides and proteins, which
have a total molecular weight of a prime number, either as a simple sum
of the molecular weights of the free aminoacids or when all water
molecules moving away are removed, multiplied by 18 (molecular weight of
water), or by other criteria. For example, we have the heptapeptide:
Methionine - Glycine - Tryptophan - Aspartic acid - threonine -
Glutamine - Cysteine. The sum of the molecular weights of these amino
acids in a free state is: 149 + 75 + 204 + 133 + 119 + 146 + 121 = 947
(prime number). Subtracting 6 x 18 = 108 (the water that is removed by
condensation), we now have 947-108 = 839 (prime number). If we multiply
the molecular weights of the amino acids to their corresponding position
in the chain, i.e., 1x149 + 2x75 + 3x204 + 4x133 + 5x119 + 6x146 + 7x121
= 3761 (prime number). The above example is as we said ''artificial'',
while in the same way we can design a lot of others, with less or more
amino acids. The point here is what happens in ''real conditions'' with
various proteins found in cells and organisms. Surely the primate to a
protein is its biological role, which is a function of the information
that exists in theDNA, and its organizational levels (primary structure,
secondary, tertiary, etc.) depending on this information. At first
glance, it may seem as a useless information compared to prime numbers
for that matter. I disagree with this point of view, especially now,
that Bioinformatics has had a great development (in collaboration with
the Proteomics course) and such studies and calculations can be made
easily and quickly. By the ''look'' in this direction, a fixed pattern
for various biological data may possibly not occur. Form of the protein
chain in space, class or subclass of proteins based on the functionality
of e.g. catalytic proteins (enzymes), etc. Some proteins that have
specificity criterion, compared with those who have not etc .. Or better
yet, something relative may occur to prime numbers in proteins with
those present in DNA, which we will see just below. It is known that a
connection is found between prime numbers and quantum physics, the
mathematics of Chaos (mathematics of unpredictability) etc. so it is not
impossible to find such a connection to Proteomics and Genomics. Besides
the innovative ideas do no harm and if anyone disagrees he can follow
what a philosopher said: ''Give me ideas to reject''. Moreover biology
is an exact science which follows the scientific method (observation,
experiment, etc.) where something wrong easily gets confuted.
\emph{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.
\selectlanguage{greek}Α \selectlanguage{english}+ \selectlanguage{greek}Τ \selectlanguage{english}+ G + C = 331 + 322 + 347 + 307 \emph{=} 1307 (2)
which is also a prime number. Easily we see that
(G + T) - (A + C) = 31 (3)
\textsubscript{11G + T - A - C = 31}
\textsubscript{14 G + 6T - 14A - 6C = 314}
\textsubscript{141 G + 59T - 141A - 59C = 3141}
\textsubscript{1415G + 585T - 1415A - 585C = 31415}
\textsubscript{14159G + 5841T - 14159A - 5841C = 314 159}
\textsubscript{Each connection gives pi = 3.14159 \ldots{} 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
\par\null
G = \(\frac{\left(A\cdot T\ +\ 10\ \cdot\ C\right)}{\left(9+C\right)}\)~ (7)
\par\null
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.
\selectlanguage{english}
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