The contingency of the connection of π (ratio of a circle’s
circumference to its diameter) with the DNA (genetic material) sounds
like impossible. In fact, they are unrelated things, and some may
characterize it as incoherence and nonsense. That is, something similar,
as if you want to connect the length of a person’s shadow with the money
he has in his pants. In the case you insist, some may guess the double
helix of DNA and its relationship to the cycle. Moreover, circular DNA
has also been found in organisms. They use to say that life and its
evolution, as well as the history of things, have a spiral shape. Let us
get things in the right order: DNA building blocks are called
nucleotides and have a specific chemical structure. Depending on the
nitrogenous base they contain, they are divided into nucleotides of
Adenine (A), Thymine (T), Guanine (G) and Cytosine (C). Their molecular
weights are 331, 322, 347 and 307, respectively. Prime are the numbers
331, 347, 307, while 322 is composite. The symbols A, T, G, C, will help
us symbolize at the same time the respective nucleotides, the respective
nitrogenous bases of the nucleotides and their molecular weights. A
mathematical formula that connects these numbers and therefore the
molecular weights of DNA nucleotides, is: G = . There is no need for
further analysis, as it results from a combination of representations
between the nucleotides.
For the record, from the sums or the differences of other
representations between the nucleotides, arise the following: Α + T + G
+ C = 1307, Α + T = 653, the graph ΑΤ – GC = 53, the (G + T) – (A + C)
= 31 etc. are all prime numbers. The latter, leads us to look for more
things. The following remains applicable:
G + 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 = 314159 etc. Table 1
The relation of each series gives us π = 3,14159… with
one, two, three etc. decimal digits multiplied by 10, 100, 1000 etc.
respectively. Furthermore, the sum of the coefficients of G and T,
respectively of A and C, is 2, 20, 200, 2000, 20000 etc. a special
connection of DNA with π . Proportionally, the approximation ofπ by Archimedes, may come to someone’s mind. On one hand, there
were sides of regular polygons inscribed, whereas here are by proportion
nucleotides. We will try to ”mathematize” various processes and see how
far we can go. Let two consecutive series e.g. ν and ν+ 1 . The coefficients of the nucleotides of the nth order are φ and x with a total result of Ζ1.
Respectively the coefficients and the result of the ν +1 series are y, ω and Ζ2. In particular:
φG + xT – φΑ – xC = Z1 (1)
yG + ωT – yΑ – ωC = Z2 (2)
Example: for the third row (ν = 3) there is φ = 141, x = 59,
Z1 = 3141. For the fourth row (ν + 1 = 4), y =
1415, ω = 585, Z2= 31415 (see table 1) etc. From the previous there is φ
+ x = 2 ⋅ 10ν−1 (3) and y + ω = 2 ⋅
10ν (4). For example, for ν = 2, φ + x = 20
and y + ω = 200 etc.
Next: From (1) φ(G – A) + x (T – C) = Z1 (5) and from
(2)
y (G – A) + ω (Τ – C) = Z2 (6).
Since G – A = 347 – 331 = 16 and T – C = 322 – 307 = 15 the
relations (5) and (6) can be written respectively:
16φ + 15x = Z1 and 16y + 15ω = Ζ2. These
last relations give us the possibility to connect all six factors of
equations (1) and (2) in an equation. Specifically, to equalize the
representations e.g. equivalent to 16. That is 16 = and to arrive at the
equation
Z2φ = Ζ1y – 15xy + 15φω (7).
Additionally, those that equate to 15, and end up in the relationship
Ζ2x = Z1ω – 16φω + 16xy (8).
From (1) and (2) by division by members easily results the (9).
By studying the initial table of series with the arithmetic
coefficients, other interesting things are being noticed, too, by which a specialist in number theory and beyond , would be pleasantly
surprised. Let us take for instance the numerical differences of the
coefficients (e.g. those corresponding to nucleotides G and T), of two
consecutive series and compare them. We see that the difference of the
coefficients in the ν + 1 series, is equal to ten times the
difference of the corresponding coefficients in the ν series,
plus twice the digit of the units of Z2 or respectively
of the coefficient of G of the ν + 1 series. Generally, y – ω
= 10 (φ – x) + 2α (10), where α is the digit of the units we mentioned.
Example between third and fourth series (ν = 3): 141 – 59 = 82
1415 – 585 = 830
830 = 10 · 82 + 2 · 5 etc. It easily follows from the relation (10) and
(4) y = 10ν + 5(φ – x) + α (11). For example,ν = 4 y = 104 + 5 (1415 – 585) + 9 ⇒ y =
14159 (coefficient G of ν + 1 = 5 series) etc. It also holds for
y = 10φ + α and Z2 = 10Z1 + α. Moreover,
interesting is the difference between the crossed products of the
coefficients, in two consecutive series. To make a long story short, the
general relation xy – φω =2 · 10ν–1α (12) is valid.
An example between the third and the fourth series: 141 · 585 = 82485,
59 · 1415 = 83485 83485 – 82485 = 1000 1000 = 2 · 102· 5 etc. Other mathematical relations can also be found and after all a
question can arise in everyone: If we know φ, x, Ζ1(previous series) and the relationship between the coefficients of the
nucleotides of the next series (as well as of other mathematical
relations), can we calculate the Ζ2 or the α? Namely the
digits of the units Ζ2 and consequently the next decimal
digit of π ? No matter how many attempts I made, no matter how
many combinations of procedures and mathematical formulas I used, it
always turned out for Ζ2 and α a representation that led
to zero. The factors were contradictory and resulted in zero results.
This forced me for a long time not to deal with the issue (to leave it
in the drawer of my office). At one point I thought (the thought
matured), that some others, based on the ideas of the essay, can find
other better and more interesting things and why not some
bio-computational approach of digits of π . On the other hand,
why should the essay have only this goal? After all, mathematicians have
found mathematical formulas and special algorithms, to calculate through
the use of computers, trillions of digits of π , and this has no
end. The essay should not be based solely on the calculation of decimal
places of π , but to show the beauty of mathematical relations
in DNA, in relation to π and where else it arises, etc.
Unfortunately, what we use to say in many cases was true for me: “I
couldn’t see the woods for the trees”. I missed something, that is
self-evident for every mathematician: “mathematicians do not do
what they do, because it is necessary and no one knows in advance if
something is needed or not ”. In the end, every cloud has a silver
lining. Along the way I found various things. Specifically, at the
beginning of the essay we saw some representations of nucleotides, which
in relation to their molecular weights give prime numbers. We can easily
and “artificially” give DNA chains with a total molecular weight of
prime numbers. Example: from the part of a DNA chain that contains
scattered but in total 29Α, 59Τ, 131G, 107C, results a molecular weight
of 106903, which is the prime number. The numbers 29, 59, 131, 107 have
been chosen to be prime numbers in this example, while the complementary
part of the chain with a set of nucleotides 29T, 59A, 131C, 107G has a
molecular weight of 106213, which is also a prime number. These sums
result from the addition of nucleotides in free form. If we remove the
water molecules (which has a molecular weight of 18) that are removed
when the nucleotides are connected together by phosphodiester chemical
bonds, in the second chain (is it transcribed?) has a molecular weight
of 100363 which is also a prime number. One might also wonder what is
happening in real conditions. Do the first numbers define properties
there? Do they leave their mark? What happens, for example, to the
genes, to their regions (introns) that do not encode amino acids, to the
regions of DNA (which are the most) and do not encode proteins, where
they encode, etc. Do they play a role in their presence or absence? Let
it be researched. Of course, the most important thing in DNA is the
sequence of nucleotides, because it determines the order of amino acids
in proteins based on the genetic code. However, the fact that the prime
numbers “have a finger in every pie” can make someone wonder whether
there is a deeper connection between the evolution of genomic
information (DNA) through natural selection, mutations, etc. and prime
numbers. In addition, in the mathematical formula (mentioned at the
beginning) and it connects the molecular weights of the nucleotides, if
we put in the place of C and G some prime number, we calculate the
product Α·Τ. If we put (in proportion of the nucleotides) in A a prime
number and in T a composite that expresses their number, the sum of the
numbers as in DNA is very often a prime number. Example: C=3, G=7 we
find Α · Τ = 54. We put A = 3 and therefore T is 18. The sum 3 + 7 + 3 +
18 = 31 (prime). If A = 2 and T = 27 does not result a prime one. The
mathematical formula is to function as a ”machine” for the production of
prime numbers. Let the prime one be investigated in relation to the
other nucleic acid, the RNA, which is also composed of nucleotides. RNA
is connected to the DNA from which it is derived, as well as to the
proteins produced by the encoded information. Who knows whether this
perspective will let us study Covid-19 in a better way? We continue with
something else amazing. The first epigenetic lesion (mutation) found in
DNA was the cytosine methylation. Methylation therein means the addition
of a methyl group, i.e. a small chemical molecule consisting of one
carbon atom and three hydrogen atoms with a total molecular weight of
15. The cytosine nucleotide gains molecular weight 307 + 15 = 322, equal
to that of thymine. From the prime number of cytosine, we come to the
composite number of thymine. And here a “confusion” arises. This
mutation in cytosine causes various phenomena (inactivation of genes, or
their activation above normal, depending on the number of methylations,
etc.) that can result in various diseases (cancers, etc.). We see the
cytosine mutation to occur on one hand at the chemical level of its
structure which is the most important, but also in the equation of the
molecular weight of its nucleotide with that of the nucleotide of
another chemical base (thymine) of DNA. Apart from the chemical change
and the change in the numbers in some proportion, they play their role
at least in the specific event. Shall we say something that may seem
strange to many? Can further or basic mathematics cure, in the future,
diseases if we associate some features with them, through some
mathematical model? Biomathematics (connection of mathematics and
biology) something similar to biochemistry and biophysics (connection of
chemistry and physics with biology respectively), have offered and will
offer more in the future, to the study of life in living organisms.
Biomathematics (biostatistics is a subset of them), in addition to the
huge contribution to science, can also offer career prospect to
mathematicians. Let us conclude the essay with something important that
was found along the way. In Table 1 (in the various series), we saw the
numerical coefficients of the nucleotides and the relationships,
developed between them. Taking into account the quotients (ratios) of
these coefficients of each series (see table 2), we observe the
following:
1/1 = 1
14/ 6 = 2,33333333…
141/ 59 = 2,389830508…
1415/ 585 = 2,418803419…
14159/ 5841 = 2,424071221…
141592/ 58408 = 2,424188467…
………… ………
…………………
2,42422678468…
Table 2
With the help of PC and for a large number of divisions (the computer
”endured” up to 307 digits of π ) the ratio of coefficients
(e.g. κ and λ) after the 12th digit, tends steadily to a very specific
number, the 2,42422678468…At first glance, this number seems
random. And I wish to protest very strongly at this. This number is very
close to the three seconds of the number Φ (golden ratio). That is \(\frac{3}{2}\) Φ =
2,427050983… A connection of DNA other than π and Φ = 1,618033988… And as we colloquially say in Greek, a
study of DNA “in pi and fi”. That means very rapidly. Nonetheless, the
approximate formula π Φ = between π and Φ, is
well-known to mathematicians. We continue: the bio-computer of π can
work even approximately with the system of equations: κ + λ = 2 ·
10ν–1 and\(\frac{κ}{λ}\approx\frac{3}{2}Φ\) or \(\frac{κ}{λ}\) = 2,42422678468… Its
calculation (the digit of its units) determines the ninth decimal digit
of π .
After all these, what could we assume as a conclusion? Is the ultimate
goal of nature to achieve the perfect circle or the perfect spiral? We
know that the logarithmic spiral is connected to Φ. Probably when this
is achieved (case of DNA formation), life is created. Life in the form
of DNA loves the perfect circle, the perfect spiral and does not reject
the golden ratio of things. Mutations that occur continuously and in
very large numbers and that lead to the removal of the stereochemical
structure of DNA (functionality and the information it contains) from
the perfection of the cycle, nature itself rejects them. It allows only
those that meet certain conditions (maintaining shape, functionality and
information) to survive, and which are generally legitimate and can
support life.