On the example of the table of contents of eight essential amino acids in 22 products, the methodology of factor analysis and
determination of the coefficient of correlation variation, calculated as the ratio of the sum of the correlation coefficients of stable
laws and laws of binary relations between amino acids to the product of the number of amino acids as influencing variables and as
dependent indicators. It is shown that this evaluation criterion depends on the composition of the products and the set of amino
acids considered. Therefore, it is proposed to make a more complete table considering the set of objects, which considers the content
of all 20 amino acids. The law of binary relations between amino acids is the sum of the exponential law and the biotechnical law of
stress excitation in the product. By correlation coefficients of individual binary relations, the ratings of amino acids as influencing
variables and as dependent indicators are performed. The correlation matrix of super strong bonds of essential amino acids with
correlation coefficients of more than 0.99 is considered, in part of which the graphs are given. By the nature of behavior, it is
proposed to classify the binary relationships between amino acids into positive, neutral and negative. Separately, the method of
rating products. The equations and graphs of rank distributions of the content of essential amino acids in products are given. The
rating of essential amino acids by dispersion of residues from the equations of binary relations and rank distributions is given.

Keywords: Amino Acids; Products; Factor Analysis; Regularities; Correlation Coefficients; Coefficient of A Correlative Variation

Under the expression correlative variation Charles Darwin
understands that the whole organization is internally connected
during growth and development, and when weak variations occur
in one part and are cumulated by natural selection, the other
parts are modified. Modifications in the structure recognized by
taxonomists for a very important, can depend solely on the laws of
variation and correlation [1]. For example, Phyto enosis [2] has at
least three fundamental properties: first, the correlative variation in
the values of parameters in time and space; second, the correlation
depends on the genotypic properties of the plant species; third, the
variation is due to phenotypic properties, as well as the cycles of
solar activity [3], the rotation of the moon around the Earth and our
planet around itself [4].

In accordance with the correlative variation of Charles Darwin,
during the growth and development of the body, significant changes
in the initial age will lead to changes in the structure and in the adult
creature. Therefore, people, taking, and thereby enhancing some
feature of, almost, probably, unintentionally modifying other parts
of the body based on the mysterious laws of correlation [1]. Any
non-investigative variation is insignificant for us. But the number
and variety of hereditary deviations in the structure, both minor
and very important in physiological terms, is infinite. All things are
things, properties and relations [5], and in the bios relations control
things and change their structures. The purpose of the article is to
show the laws and regularities of the correlative variation of the
content of eight essential amino acids in a set of 22 products.

The Concept of Correlative Variation

Variability [1] is usually associated with the living conditions
that the species has been subjected to for several successive
generations. In General, according to Darwin, there are two factors:
the nature of the body (most important factor) and the properties
of existing conditions [1]. Thus, we adopted the basic hypothesis
that typing, and the classification has no effect on biotech based on
mastery of life laws. Therefore, the variation (the set of deviations
from the Darwin correlation) depends on the human factor, i.e. on
the quality of measurements of soil properties and plants [4,6].
Soil according to V.V. Dokuchaev [7] is a living organism. Therefore,
the principle of Darwin’s correlative variation should provide high
adequacy of the revealed regularities [6]. Similarly, a priori we will
consider experiments with essential amino acids [8] to measure
their concentration in different types of products for humans and
animals highly correlative.

From the concept of correlative variation of Charles Darwin,
which was not understood by mathematicians and was not
developed by biologists, it clearly follows that in other conditions
of the habitat other combinations of values of factors may be
stronger (Darwin calls factors hereditary deviations). Therefore,
weak factor connections may be stronger for other combinations
of the studied objects. As a result, there is a mathematical tool [9-
12] (identification method) for comparison of different natural
and artificial (technical) objects [13]. The coefficient of correlative
variation is considered for many factors of the physical object of
study, that is, biological, chemical, technological, socio-economic,
etc. It is equal to the ratio of the total sum of the correlation
coefficients to the square of the number of factors for the complete
table model (or to the product of the number of factors and). The
type of the system under study does not affect this criterion, and the
correlation variation depends entirely on the internal properties of
the system under study. The coefficient of correlative variation is
calculated by the formula

where

K – the coefficient of correlative variation of the set of factors
or parameters characterizing the system under study,

ΣΣr – the total sum of the correlation coefficients for the
rows and columns of the correlation matrix of the relationships
between the factors,

N – number of factors to consider in the symmetric table,

N_{x}, N_{y} – the number of factors on the axes x and y .

Functional connectivity is a universal property of matter. For
example, internal correlation variation is observed in the results
of agrochemical analysis of soil samples [6], as all agrochemical
parameters are measured on the same sample. Sampling sites
do not affect the internal connectivity of biochemical and other
reactions, that is, the same interactions between chemical elements
and their compounds are observed on Earth. Such a community is
called an ecosystem [2] or a biosphere superposition. The strongest
correlative variation over 0.999 is observed in the set of genes [13-
15]. Slightly less, but more than 0.99, as will be shown in this article,
such a variation exists in the group of essential amino acids.

These are essential amino acids for animals that cannot be
synthesized in the body, in particular, human. Therefore, their
intake from food is necessary (Table 1).

Table 1: The content of essential amino acids in products [1] (grams per 100 grams of product), We have included in the list of
products №22 «Shiitake mushrooms».

Rating of Influencing and Dependent Factors

To determine the coefficient of correlative variation of nine
amino acids among 22 types of products it is necessary to conduct
a factor analysis [9]. Due to the absence of a measured value of
glycine content in one cell of Table 1 in the row Shiitaki mushrooms,
factor analysis was first carried out [9, pp. 82-83, table. 3.30] nine
factors and 21 products. The coefficient of correlative variation was
equal to 0.9985. All binary 92– 9 = 72 relations are characterized by
the exponential law. Table 2 shows the correlation matrix of binary
relationships and the rating of eight factors excluding glycine for 22
products according to Table 1.

Table 2: Correlation matrix of factor analysis without glycine and rating of factors in identification by the exponential law.

Table 3: Correlation matrix of factor analysis without glycine and rating of factors in identification by exponential and biotechnical
law.

Table 3.1:

The coefficient of correlative variation is 0.9727, which is
significantly less than 0.9985. In the future, it turned out that in
addition to the indicative law, the biotechnical law [4,6,9-15] of
the stress excitation of amino acids depending on each other is
additionally considered (Table 3). With the coefficient of correlative
variation 0.9890 in the first place among the influencing variables
was lysine, and among the indicators -phenylalanine. Thus, the
correlative variation is very sensitive to the composition of amino
acids and products. This fact in the future will reveal the rational
compositions, structures and functions of amino acids in different
systems under study.

The Law of the Relationship Between Amino Acids

It is expressed by an equation of the form

where y- amino acid content in the product as an indicator (g
per 100 g of product),

x- amino acid content of the product as an influencing variable
(g per 100 g of product),

a_{1.....} a_{6} - the parameters of the model (2) taking the numerical
values in the course of structural-parametric identification in the
software environment CurveExpert-1.40. Formula (2) shows three
types of stress-induced amino acids under the influence of each
other: positive, neutral and negative. Neutral type appears only
without shiitake mushrooms, that is, when the amino acid content
changes from 0 to 2 (maximum 2.009 for beef). The maximum
concentrations of nine amino acids without shiitaki mushrooms are
in two products-beef and chicken meat. In the amino acid content
range from 2 to 7 in Table 1 there are no types of products (except
mushrooms). Therefore, it is necessary to add new products to the
list and Table 1.

In the concentration range from 0 to 7, two types of behavior
appear:

positive behavior, with a positive sign in front of the
second component of the formula (2), when with increasing content
of the influencing amino acid, the content of the dependent amino
acid increases according to the biotechnical law of stress excitation;

negative behavior, with a negative sign, when the content
of the dependent amino acid is inhibited from the action of the
wagging amino acid. These two types provide optima for the
interaction of essential amino acids.

Table 4: Correlation matrix of super strong binary relations by exponential law at the level of adequacy r ≥ 0,99.

Table 5: Correlation matrix superpowered binary relations exponential and biotech law, when the correlation level r ≥ 0,99 .

From Table 2, choose a binary relationship with a correlation
coefficient of at least 0.99 (Table 4). The neutral behavior of amino
acids receives only 15 binary bonds at the level of superstrong
adequacy (or 100 15 / 72 = 20.83%). We will do the same with the
data in Table 3. Table 5 shows a significant increase in the number of
super strong links, 33% or 45.83%. The matrix remained completeeight
rows and columns. However, the behavior of biological objects
is characterized, in addition to the trend (2) vibrational adaptation
[6,12,14,15]. To identify the wavelet signals from the amino acid
behavior under mutual influence, it is necessary to significantly
expand Table 1 also with interchangeable amino acids. Especially
it is necessary to pay attention to the types of products that give a
concentration of 2 to 7.

Binary Relation Graphs

The effect of each essential amino acid on the concentration of
other amino acids is shown in four graphs, which are arranged in
figures in descending correlation coefficient. Of the 72 graphs in
the article shows a total of 32 graphics (Figures 1-8). They provide
a visual representation of the variations of formula (2). The second
component of the general model (2) shows a different level of
adaptability (positive or negative) dependent on the influencing
amino acid by the coefficient of adaptability.

Figure 1: Graphs of the effect of leucine on other essential amino acids (in the upper right corner: S - variance; r - correlation
coefficient).

Figure 2: Graphs of the effect of isoleucine on other essential amino acids.

Figure 3: Graphs of the effect of histidine on other essential amino acids.

Figure 4: Graphs of the effect of tyrosine on other essential amino acids.

Figure 5: Charts the influence of the lysine to other essential amino acids.

Figure 6: Graphs of valine effect on other essential amino acids.

Figure 7: Graphs of the effect of methionine on other essential amino acids.

Figure 8: Charts the influence of phenylalanine to other essential amino acids.

On the charts, the positive behavior of essential amino acids
is shown as a convex curve, and the negative behavior is shown
as a concave curve. On the effect of histidine and methionine it is
possible to estimate the minimum interval of the concentration of
essential amino acids with neutral type. As can be seen from the
graphs in Figure 3, the neutral behavior type is from 0 to 0.5. The
convexity or concavity of the graph has a different length along the
abscissa axis. It is obvious that the addition of the list from Table
1 with additional products having amino acid concentration in the
range from 2 to 7, will allow to specify the parameters of the model
(2) and add a wave function. However, in our opinion, the nature
(design) (2) of the expansion of the list of products will not change.
Graphs of the effect of lysine on other amino acids have a concave
appearance, which relates them to the negative effect. At the same
time, there is a steep rise of curves, more than five times after the
concentration of lysine 2.5-2.8. Figure 7 shows the multidirectional
effect of methionine on other amino acids along the different length
of the concave or convex part. For example, the positive effect of
phenylalanine on the change of isoleucine is observed in a short
period of abscissa from 0.3 to 1.3. But the effect of phenylalanine on
the change of leucine is observed throughout the axis of abscissa.

Parameters of a Two-Member Model

Table 6: Parameters of the laws of mutual influence of essential amino acids.

Table 7: Behavior of essential amino acids.

Model parameters (2) are given in Table 6 The positive form of
the equation (2) is used in the recording. Then the negative type is
easily determined by the negative sign before the model parameter
a3. Then you can classify (Table 7) three types of behavior of
essential amino acids (code: 1 – positive; 0 – neutral; -1 – negative).
Apparently, for the most complete amino acid system, the total sum
of codes will approach zero. As a variable on the positive effect in
the first place – phenylalanine, and among the dependent indicators
– lysine.

The Quality of the Source Data and Ratings of Products

A tabular model is a good-quality and relatively complete table
of input data for statistical modeling by identification of stable laws
and regularities. The quality factor is understood as the accuracy of
the numbers, the primacy of the indicators (the factor analysis is not
initially permitted secondary received, calculations, parameters),
consistency of the description of the object of the studies consider
many parameters. As a result, we get the distribution of places in
descending order (Table 8).

Table 8: Rating of products by content of essential amino acids.

Let’s check the initial data of Table 1 for the quality factor.

Any of the factors is the vector orientation and the two possible
behaviors:

better more on the vector better worse, the rank is given
to the maximum, and the ranking is performed in descending order
of the factor values;

it is better to lower, so the rank is given to a minimum,
and the ranking is performed by increasing the values of the factor.

From the hypothesis- the greater the content of any essential
amino acid in any product, the better - the first option of a vector of
behavior is accepted. We range each amino acid from Table 1 in Excel
in the program RANK. The function =RANK (Е3; Е$3: Е$24;0) taken
notation: Е – column identifier; Е3, Е$3 – the first row; E$24 – the
last row of the Table 1; 0 1 – ranking in descending (0) or ascending
(1). Three places in the ranking took products: mushrooms, beef
and chicken. The rating in Table 8 is determined in places. The best
theoretical first place is obtained if. Then it turns out that shiitake
mushrooms in a variety of products according to the Table 1 is theoretically possible in the first place. For the second place the
sum of ranks is 12. Next, we can take places for the explanatory
variable, and the sum of ranks from Table 8 for the indicator.
After identification of the general trend formula [13], we
obtained (Figure 9) formula

Figure 9: Product rating charts from table 1 data.

The remnants after (3) show that wave function is possible
in addition to the trend. Half amplitude reaches a share of 100 ×
17.3524 / 85 = 20.41%. In the article we do not consider vibrational
adaptation, as we need a table for all 20 amino acids. Each of the
eight amino acids will be considered separately.

Figure 10: Product rating charts from table 1 data.

For statistical modeling, the ranks R must start from zero, with the rank distributions subject to the exponential law (growth or
death). After the identification of the generalized trend [13] are obtained (Figure 10) formulas:

rank distribution of leucine

rank distribution of isoleucine

rank distribution of histidine

rank distribution of tyrosine

ranking distribution of lysine

rank distribution of valine

rank distribution of methionine

rank distribution of phenylalanine

The sum of the squares of deviations from the equations of binary and unary (by ranks) relations between eight essential amino acids
is written in the dispersion matrix (Table 9). As the influencing variable on the first place on minimum of the sum of dispersions there was
isoleucine, and among dependent indicators – methionine. The average variance for all 8^{2} = 64 cells of the matrix is D = 0.1231.

Table 9: The variance matrix of the residuals after the full factor analysis and ranking of factors when identifying significant and
biotech law.

We have extended the principle of correlative variation not
only to Charles Darwin organisms, but also to populations (in the
article population of eight amino acids) and even to any biological,
biotechnical and technical systems [13]. This principle allows to
compare heterogeneous systems on one or some set of factors by
functional connectivity. The coefficient of correlative variation, as a
generalized criterion for comparing different sets of homogeneous biological objects, gets a very high value. For example, populations
of genes [14,15] obtained the correlative coefficient of variation not
less than 0.9999. In the example of this article, the level of adequacy
for the set of eight essential amino acids is not less than 0.99. This
makes it possible in the future to create the most complete table of
contents and other indicators for a system of 20 amino acids and
hundreds of objects, including products. Functional connectivity
between essential amino acids was super strong and it is subject
to a simple formula of two-term trend containing exponential and
biotechnical laws. The absence of the second term determines
the neutral type of behavior, and signs in the presence of the
second member characterize the positive (+) or negative (-) type
of behavior of amino acids in the studied system of products. To
identify the effect of oscillatory adaptation of essential amino acids
to each other in some sets of products need more accurate (with
measurement error, less than an order of magnitude) data.