Shcherbak’s arithmetic and Boulay’s arithmetic: the proline hypothesis.
1.
Introduction.
In
this paper, the author proposes a new hypothesis on the structure of proline.
This hypothesis confirms the Shcherbak’s arithmetic
and feature unusual arrangements in the counting of the various entities
(nucleons, atoms, etc.) which make up the amino acids.
1.1. Proline specificity.
The
proline is the only amino acid which has a very particular chemical structure.
The base and the radical of this amino acid are fixed two times. So, the proline has a
base which is different of all other amino acids with a lost hydrogen atom.
Common structure of
each AA (here SER) 
Particular proline structure 


Radical zone and base zone are fixed one time 
Radical zone and base zone are fixed two
times 
In his arithmetic, Shcherbak conserves a hydrogen atom in the counting of the
proline base but he removes this hydrogen from the radical. In Boulay’s arithmetic, this hydrogen atom is also conserved
in the base but it’s not removed from the radical. Without these two special
counting, the arithmetic of Shcherback and of Boulay would
be completely destroyed! Without these two special counting, the
very numerous and organized phenomena presented by Shcherback and Boulay disappear.
These respectively presented phenomena are too much numerous and too much
structured to be destroyed by an apparent contradiction of chemical
arrangement.
1.2. “The
proline hypothesis” depiction.
“The proline hypothesis” 
The author,
JeanYves Boulay, thus proposes a hypothesis in order to preserve these two arithmetic. It must occur
a “singularity” with the second junction (specific junction to the proline)
between the base and the radical. The author proposes a hypothesis that a
neutron could behave as a proton. In
this event, the counting of the particles of the carbon where is located this
singularity gives 5 neutrons and 7 protons including 6 protons to be counted
for the radical and 1 proton for the base. This possibility also lets suppose
a new chemical (or physical) property of the nucleons which can to be
interesting to study. 

Depiction of the proline structure in a new proposed hypothesis:
Special particles
counting of the proline (“the proline hypothesis”) 


Hydrogen 



Carbon 


Nitrogen 


Oxygen 




Entities 
Complete AA 
Radical zone 
Base zone 
Protons 
63 
24 
39 
Neutrons 
52 
17 
35 
Nucleons 
115 
41 
74 
Atoms 
18 
9 
9 
This new structure of the
proline is perfectly in agreement with the phenomena presented by Shcherbak and Boulay [3]. The
presented following phenomena are in accord with this new counting system: the
proline is constituted by 18 atoms, 115 nucleons, 63 protons and 52 neutrons.
2.1. Shcherbak’s arithmetic.
In an article [1], Shcherbak
have described two sets of amino acids which have a nucleons number multiple of
the prime number 37.
The two Shcherbak’s sets
which have a nucleons number multiple of the prime number 37 


SET 1 

SET 2 







Coded (amino acids) 
Nucleons in base 
Nucleons in radical 
Nucleons in complete AA 

Coded (amino acids) 
Nucleons in base 
Nucleons in radical 
Nucleons in complete AA 


GLY 
74 
1 
75 

SER 
74 
31 
105 



74 
15 
89 

CYS 
74 
47 
121 


SER 
74 
31 
105 

LEU 
74 
57 
131 


PRO 
74 
41 
115 

ILE 
74 
57 
131 


VAL 
74 
43 
117 

ASN 
74 
58 
132 


THR 
74 
45 
119 

ASP 
74 
59 
133 


LEU 
74 
57 
131 

GLN 
74 
72 
146 


ARG 
74 
100 
174 

LYS 
74 
72 
146 







GLU 
74 
73 
147 







MET 
74 
75 
149 







HIS 
74 
81 
155 







PHE 
74 
91 
165 







ARG 
74 
100 
174 







TYR 
74 
107 
181 







TRP 
74 
130 
204 


8 
592 
333 
925 

15 
1110 
1110 
2220 



4 x 4 x 37 
3 x 3 x 37 
5 x 5 x 37 


2 x 3 x 5 x 37 
2 x 3 x 5 x 37 
4 x 3 x 5 x 37 












In the first
set are included the height coded (amino acids) which are coded four times by
the codons (triplets) with the two first identical DNA bases. In the second set
are included the alls coded (amino acids) which are coded three, two or one
times by the codons (triplets) with the two first identical DNA bases.
The nucleons
numbers of these two sets are multiple of the prime number 37: 25 times 37 for
the first set and 60 times 37 for the second set. These two counting are not
the totality of nucleons of the 64 coded but only the sub totals of the amino
acids counted one time for each set. Nevertheless the total nucleons number for
the 64 coded is also multiple of 37.
2.2. Boulay’s rebel group.
Boulay observation (The numeric connections of the genetic code
20012005^{© }^{ }on the Web only) [3]:
Systematically, it seems that the coded are
identical if the final base of the codon is either A or G or if this
base is either T or C. This except for a named group the rebel
group :
The rebel group set apart, codons code for the same
coded if and only if their last base is either A or G or
their last base is either T or C. 
For the first Shcherbak’s set, the total nucleons number is, of course,
multiple of 37 because the AA are all represented four times (2 times + 2
times):
925 x 4 = 3700 = 100 x 37.
For the second set, the AA are
not alls represented two times but the total nucleons number is nevertheless
multiple of 37. Because? Of cause to the special configuration of the rebel
group! ILE is represented 3 times, MET
and TRP 1 time only. The arrangement of the nucleons numbers in these three AA
preserves the arithmetical phenomena of multiples of prime number 37:
Nucleons in the set 2 of AA 

2220 
60 x 
37 
All AA are represented 2
times (except…) 
x 2 = 
4440 
120 x 
37 
…ILE is represented 3 times
Þ (+ 1 time) 
+ 131 
}  222 
 6 x 
37 
MET is represented 1 only
time Þ ( 1 time) 
 149 

TRP is represented 1 time
only Þ ( 1 time) 
204 

STOP is
represented 3 times (+ 1 time) 
+ 0 



total nucleons number in
set 2 

4218 
114 x 
37 
total nucleons number in the two Shcherbak’s
sets 


SET 1 

SET 2 



Coded (amino acids) 
Nucleons in complete AA 
Number of coded for the AA 
Total nucleons number 

Coded (amino acids) 
Nucleons in complete AA 
Number of coded for the AA 
Total nucleons number 


GLY 
75 
2 + 2 
300 

(STOP) 
 
(2 + 1) 
(0) 



89 
2 + 2 
356 

SER 
105 
2 
210 


SER 
105 
2 + 2 
420 

CYS 
121 
2 
242 


PRO 
115 
2 + 2 
460 

LEU 
131 
2 
262 


VAL 
117 
2 + 2 
468 

ILE 
131 
2 + 1 
393 


THR 
119 
2 + 2 
476 

ASN 
132 
2 
264 


LEU 
131 
2 + 2 
524 

ASP 
133 
2 
266 


ARG 
174 
2 + 2 
696 

GLN 
146 
2 
292 







LYS 
146 
2 
292 







GLU 
147 
2 
294 







MET 
149 
2 – 1 
149 







HIS 
155 
2 
310 







PHE 
165 
2 
330 







ARG 
174 
2 
348 







TYR 
181 
2 
362 







TRP 
204 
2 – 1 
204 


8 
925 
32 
3700 

16 
2220 
32 
4218 



5 x 5 x 37 
4 x 8 
10 x 10 x 37 



(2 x 16) + 2  2 
114 x 37 












2.3. Others similarity between Shcherbak’s
observations and Boulay’s observations.
The numbers of the different
qualities of particles in each base of amino acid and in the totality of the 64
coded are multiple of the same prime numbers: 37 for the nucleons, 13
for the protons and 7 for the neutrons.
particles 
in the base of each AA 
in all amino acids 
nucleons number 
74 = 37 x 2 
7918 = 37 x 214 
protons number 
39 = 13 x 3 
4264 = 13 x 318 
neutrons number 
35 = 7 x 5 
3654 = 7 x 522 
With or without the rebel
group, the numbers of the different qualities of nucleons are multiple of the
same prime numbers: 13 for the protons and 7 for the neutrons.
protons number in all amino acids 
protons number in the rebel group only 
protons number in all AA without the rebel group 
4264 = 13 x 328 
260 = 13 x 4 x 5 
4004 = 13 x 7 x 11 x 2 x 2 
neutrons number in all amino acids 
neutrons number in the rebel group only 
neutrons number in all AA without the rebel group 
3654 = 7 x 522 
224 = 7 x 4 x 8 
3430 = 7 x 7 x 7 x 5 x 2 
The same
phenomenon appears with the counting of Petoukhov’s
modules* [4]:
modules number in all amino acids 
modules number in the rebel group 
modules number in all AA without the rebel group 
544 = 17 x 32 
34 = 17 x 2 
510 = 17 x 30 
Also:
counting of the modules liaisons in all amino acids 
counting of the modules liaisons in the rebel group 
counting of the modules liaisons in all AA without
the rebel group 
495 = 11 x 45 
33 = 11 x 3 
462 = 11 x 42 
*The Petoukhov’s
structure is a original system describing the molecular structure: 
For example, atom of nitrogen will be represented by
number 7. If one or several atoms of hydrogen are joined to some nonhydrogen
atom, we shall add protons of hydrogen to number of protons of nonhydrogen
atom always. Such group of the nonhydrogen atom with its adjoined hydrogen
atoms (i.e., “protonated” nonhydrogen atom) will
be denote by sum of protons of their nucleuses. For example, amino group NH_{2}
will be denoted by number 9 which is sum of 7 protons of nitrogen atom and 2
protons of two hydrogen atoms. In such schemes the atoms of hydrogen are not
represented independently at all, and their presence at a molecule simply
increases general number of protons of the atoms, connected to them directly. 
*Extrait
of : S.V.Petoukhov “Genetic Code and the Ancient Chinese “Book Of Changes””
, “Symmetry: Culture and Science”, 1999, vol.10, № 34, p. 211 
For more details on the Petoukhov’s structure please consult: Petoukhov’s modules 
3. Similarity between atoms
numbers and nucleons numbers
The total number of atoms in
the 64 coded is equal to: 1147 = 37 x 31
Without the rebel group, the total
number of atoms in the 64 coded is equal to: 1078 = 7 x 7 x 22
So, the atoms number and the
nucleons number are multiple of the same prime numbers in the different
counting.
Similarity between atoms numbers and nucleons
numbers 

counting depiction 
Atoms number 
Nucleons number 
total number in the 64 coded 
1147 = 
7918 = 
31
x 37 
214
x 37 

total number in the 64 coded without the rebel group 
1078 = 
7434 = 
22
x 7 x 7 
1062
x 7 
So, without the rebel group, the numbers of atoms, nucleons, protons and neutrons are
all multiple of the prime
number 7:
multiples of prime
number 7 in according to different entities 

entities 
entities numbers 

atoms 
1078 = 
22 x 7^{2} 
nucleons 
7434 = 
1062 x 7 
protons 
4004 = 
572 x 7 
neutrons 
3430 = 
10 x 7^{3} 
4. Tables of counting of the different
entities
Counting of different entities in each amino
acid. These counting are in according with the special proline depiction which
is presented in the proline hypothesis:
counting of different
entities in each amino acid 

entities counting in the complete amino acids (base
+ radical which are united) 
entities counting in the amino acids according to
the counting system of apartheid* 




*entities counting in the amino acids in according to
the counting system of apartheid: this counting system is presented in the
following chapter (just now)
counting of different
entities in the 64 coded 

entities counting in the complete amino acids (base
+ radical which are united) 
entities counting in the amino acids according to
the counting system of apartheid* 




5. Study of the counting of nucleons by physical
apartheid of the radical and the base of the amino acid.
5.1. Depiction of the counting of nucleons by
physical apartheid.
In this
counting system, the radical and the base of each amino acid are separated in
two molecules and the two molecules are closed by a hydrogen atom. By this new
counting system, only the protons numbers are different. This is just a
theoretical (algebraic) separation bat not a real physical apartheid. So, for
proline, is preserved a singular counting of neutrons number in the carbon atom
which is affixed to the nitrogen atom of the base: 5 neutrons only.
Detail of apartheid system (common and special
configurations) 

For each amino acid: + one hydrogen in the base and + one hydrogen in the radical 




For example : SER = 56 protons 
New counting : SER = 58 protons 

Glycine Particularity 




GLY = 40 
Glycine is separated also 
New counting GLY = 42 

Proline Particularity 




PRO = 63 
only + one hydrogen 
New counting PRO = 64 

By this new counting
system, the phenomena of multiples of prime number 7 are much amplified.
5.2. Counting of nucleons by physical apartheid
system.
Counting of
the different particles in the genetic code table:
Counting of protons, neutrons
and nucleons in the
genetic code table by apartheid system counting 


A 
G 
T 
C 


A




G



T



C



Neutrons number 
Protons number 
Nucleons number 

With the rebel group: 

3654 
4382 
8036 

7
x 522 
7 x 626 
7
x 7 x 164 

Rebel
group only:


224 
266 
490 

7
x 32 
7 x 38 
7
x 7 x 10 

Without the rebel group: 

3430 
4116 
7546 

7
x 7 x 7 x 10 
7 x 7 x 7 x 12 
7
x 7 x 7 x 22 

5.3. Arithmetic results.
Counting of
nucleons in this apartheid system reveals surprising results: the totals are
all multiples of 7 with or without the rebel group. Also, without the rebel
group, the totals are multiples of 7 x 7 x 7 (so 7^{3}):
With or
without the rebel group, the total protons number is multiple of the prime
number 7:
with the rebel group: 
4382 protons 
= 7 x
626 
the rebel group
only: 
266 protons 
= 7 x 38 
without the
rebel group: 
4116 protons 
= 7 x 7 x 7 x 12 
With or without the rebel
group, the total neutrons number is multiple of the prime number 7:
with the rebel group: 
3654 neutrons 
= 7 x
522 
the rebel group
only: 
224 neutrons 
= 7 x 32 
without the
rebel group: 
3430 neutrons 
= 7 x 7 x 7 x 10 
With or without the rebel
group, the total nucleons number is multiple of the number 49 (so
7^{2}). Without the rebel group
the total protons numbers and the total neutrons numbers are multiple of the
number 343 so multiple of 7^{3 }(so 7 x 7 x 7)!
with the rebel group: 
4382 protons 
= 7 x 626 
3654 neutrons 
= 7 x 522 

8036 nucleons 
= 7 x 7 x 164 
the rebel group only: 
266 protons 
= 7 x 38 
224 neutrons 
= 7 x 32 

490 nucleons 
= 7 x 7 x 10 
without the rebel group: 
4116 protons 
= 7 x
7 x 7 x 12 
3430 neutrons 
= 7 x 7 x 7 x 10 

7546 nucleons 
= 7 x 7 x 7 x 22 
5.4. Depiction of symmetrical and identical
configurations in genetic code table.
In the genetic code table
appear symmetrical and identical configurations of multiples of the prime
number 7 for the totality of the nucleons, for the protons only and for the
neutrons only:
In the following descriptions, this genetic code table is represented
in a compressed version: 

A box is the total sum (nucleons number) of 4 coded with two identical
first DNA bases. 

Symmetrical and identical configurations (without the
rebel group)


Total
nucleons numbers



2
x 11 x 7 x 7 x 7 


6
x 11 x 7 x 7 

8
x 11 x 7 x 7 





Protons
numbers only

Neutrons
numbers only


12
x 7 x 7
x 7 
10
x 7 x 7 x 7 

6
x 6 x 7 x 7 
6
x 8 x 7 x 7 
6
x 5 x 7 x 7 
5
x 8 x 7 x 7 






Symmetrical and identical configurations (with the rebel
group)


Total
nucleons numbers


4
x 19 x 7 x 7 

4
x 22 x 7 x 7 


4
x 41 x 7 x 7 


Rebel group only: 10 x 7 x 7 

Protons
numbers only

Neutrons
numbers only


10
x 29 x 7 
6
x 8 x 7 x 7 
11
x 22 x 7 
5
x 8 x 7 x 7 






626
x 7 
522
x 7 

Rebel group only: 38 x 7 
Rebel group only: 32 x 7 

Also, without the rebel group, and by counting system
of physical apartheid of the radical and the base of the amino acid, the total number
of atoms in the 64 coded is equal to:
1190 = 170 x 7
6. Arithmetical conclusions.
So, without the rebel group, the numbers of atoms, nucleons, protons and neutrons are all
multiples of the prime number
7. This,
with and without the counting system of physical apartheid. with the counting
system of physical apartheid, the phenomena of multiples of 7 are very
amplified.
Multiples of prime number 7 according to different
entities:
counting configuration 
entities 
entities numbers 

without the rebel group (base
+ radical which are united) 
atoms 
1078 = 
22 x 7
x 7 
nucleons 
7434 = 
1062 x 7 

protons 
4004 = 
572 x 7 

neutrons 
3430 = 
10 x 7 x 7 x7 
counting configuration 
entities 
entities numbers 

without the rebel group
and by counting system of physical apartheid 
atoms 
1190 = 
170 x 7 
nucleons 
7546 = 
22 x 7 x 7 x7 

protons 
4116 = 
12 x 7 x 7 x7 

neutrons 
3430 = 
10 x 7 x 7 x7 
The author suggests that these results are
not due to chance and that the genetic code has not formed at random but
responds to numerical constraints. Also, the proposed hypothesis on the atomic
composition of the amino acid proline is a serious investigation to track in
the study of the genetic code and in the field of physics.
Bibliographies :
[1] Shcherbak, V. I.
Sixtyfour triplets and 20 canonical amino acids of genetic code: the
arithmetical regularities. Part II //J. theor.
Biol., v. 166, p. 475477, 1994)
[2] S.V.Petoukhov
“Genetic Code and the Ancient Chinese
“Book Of Changes””, “Symmetry: Culture and Science”,
1999, vol.10, № 34, p. 211
[3] The numeric connections of the
genetic code  JeanYves Boulay 20012006^{© }(on the Web only)
[4] Connections between Sergei PETOUKHOV searches and JeanYves BOULAY searches 2005^{© }(on the Web only)^{}
Shcherbak’s
arithmetic and Boulay’s arithmetic: the proline
hypothesis. JeanYves Boulay 20042010^{©}
To return to: 
THE NUMERIC CONNECTIONS
OF THE GENETIC CODE JeanYves BOULAY 2001 2010©