Abstract (English):
Introduction. The existing diversity of plant raw materials and products predetermine the prospects of studying their potential as sources of pectin substances. However all current classifications are either fragmented or inconsistent. Study objects and methods. Our theoretical ivestigation aimed to develop an adequate classification for all taxa of plant origin, as well as their tissues and derivatives as pectin-containing materials. We developed criteria for assessing transformation potential of the protopectin complex based on the mass fractions of biologically active non-uronide components, native water-soluble pectin, the protopectin complex, and pectin substances. Individual boundary conditions were based on individual pectin potential, protopectin fragmentation potential, and pectin isolation potential. Results and discussion. Based on the boundary conditions, we defined an universal criterion space that included a set of points M in the coordinates expressed by three main criteria. According to individual boundary conditions, the criterion space was divided, or zoned, into four domains corresponding to protopectin fragmentation potential. They were characterized by: 1) lack of pectin potential, 2) ineffective protopectin fragmentation, 3) ineffective isolation of fragmentation products, and 4) effective isolation. Finally, we developed a generalized algorithm to determine the location of points M[μ1, μ2 , μ3 ] in the zoned criterion space, characterizing the plant tissue. Conclusion. Our approach can be used to assess any plant tissue for its protopectin transformation potential, which determines the technological influence on its pectin potential. This approach is universal, i.e., applicable to both plant tissue and its derivatives.

Keywords:
Protopectin complex, potential, transformation, evaluation system, criterion space

INTRODUCTION
Food technology is currently striving to maximize
the potential of raw materials and use new, nontraditional
sources of essential nutraceuticals and
food components with biological (antioxidants,
enterosorbents, etc.) and/or technological (thickeners,
stabilizers, etc.) functional activity [1, 2]. The most
promising way to achieve that is a biotechnological
approach that makes use of both living cultures of
microorganisms and isolated enzyme systems. When
using isolated enzyme systems, this approach involves a
multiple stage fragmentation of a native supramolecular
complex of plant and/or animal cell walls into target
components with a wide range of physicochemical and/
or technological properties [3–5].
One of the methods within this approach is to
activate the potential of a multicomponent polymer
matrix of cell walls and intercellular spaces. This
method has a limited use in processing agricultural
raw materials. It mainly consists in partial or complete

degradation (depolymerization) of its individual
components to change the consistency or transparency
of the final product, or to clear it of degradation products
and improve its sensory characteristics. Most certainly,
a targeted use of this polymer matrix is complicated by
its highly heterogeneous components, a system of bonds
between them, and highly entangled polymer chains [6].
Moreover, the heterogeneity of individual matrix
components is a serious obstacle to controlling their
properties during extraction [7, 8].
Pectin substances are among major carbohydrate
biopolymers that have a wide variety of functional and
technological characteristics [9, 10]. In a plant cell,
they are represented by two main fractions – native
water-soluble pectin and a native water-insoluble
protopectin complex. The last one is the most valuable
for transformation due to its molecular structure and
composition [9].
The structure of cell walls in almost all terrestrial
plants [6, 11, 12] makes them a potentially good resource
for the industrial production of pectin. However, it is
difficult to implement. Since the protopectin complex is
a branched supramolecular structure incorporated into
the cell wall, its transformation is mainly fragmentation
into water-soluble polymers (soluble pectin). In addition,
mass fractions of pectin substances and the protopectin
complex may depend on the type, grade, and purpose of
raw materials, their structure and phase of development,
soil and weather conditions for their vegetation, as
well as localization, duration and storage conditions,
processing intensity, etc. [10, 13]. In this regard, the
choice of a plant as a pectin-containing material should
be determined by the purpose of its use.
Raw materials can be classified according to the
size of their pectin potential – “high”, “medium,” and
“small” (“low”, “insignificant”) [9, 10, 14]. The only
fundamental approach to pectin production was offered
by Donchenko in [15] and supplemented by Rodionova
et al. in [19, 20] (works [16–18] are actualy based on
[15]). Although this approach is rather fragmented, it
can be used as a basis for developing a universal system
that takes into account the native pectin potential
of plant tissue.
The protopectin complex is a key object whose
fragmentation enables us to use the biomass of a plant
material as a source of pectin substances. Due to the
presence of certain plant organisms, mainly a natively
soluble fraction of pectin, biomass can be attributed
to potential sources of pectin. On the other hand, the
biomass of certain taxonomic elements may contain a
small amount of pectin, which makes its use ineffective.
Therefore, we found it relevant to develop a clearcut
classification of plant bio-resources into groups to
determine the prospects of their use as pectin-containing
raw materials.
In this regard, we aimed to develop a system of
criteria for assessing the transformation potential of
native complexes of plant carbohydrate biopolymers
exemplified by pectin. To achieve this aim, we set the
following objectives:
– working out criteria to assess the transformation
potential of native plant biopolymers and the concept of
their applicability, and
– developing a system of boundary conditions and
an universal algorithm for classifying plant materials
according to the transformation potential of their native
pectin components.
STUDY OBJECTS AND METHODS
According to existing data, all plant materials can be
classified into four main groups, namely:
– bio-resources with sufficient potential for protopectin
fragmentation and subsequent isolation of its products as
independent substances;
– bio-resources with sufficient potential for protopectin
fragmentation, but with insufficient potential for
isolation of its products;
– bio-resources with insufficient potential for
protopectin fragmentation, but with sufficient potential
for natively soluble pectin;
– bio-resources with no pectin potential.
On the one hand, this differentiation involves
unifying plant characteristics and reducing them to
certain generalized values. On the other hand, it involves
dividing the domain of generalized values into four fixed
zones. As we know, a universal tool for unifying an
arbitrary set of source factors is a range of anonymized
criteria reducible to a certain system with the use of
boundary conditions [21, 22]. Thus, we can apply a
criteria-based approach to fulfilling our objectives.
To be able to scale the criteria to determine clear
boundary conditions, we used Harrington’s individual
desirability function in its canonical form [23]:
e (bi0 bi1 i )
di e = − − + ⋅ϕ (1)
where di is the dimensionless value of Harrington’s
individual desirability function; bi0 is the constant; bi1 is
the coefficient; and φi is the dimensionless operator of
Harrington’s individual desirability function.
We introduced the first and second individual criteria
for protopectin fragmentation potential among the main
criteria to assess the native pectin potential.
Let us begin with the first criterion. According to
[7, 8], the presence of pectin in the tissue or a certain
amount of protopectin in the cell wall matrix is not
sufficient for assessing the native pectin potential of
plant tissue. The tissues of many plant organisms also
contain a significant amount of organic and mineral
components with valuable vitamins and antioxidant
activity, pronounced aroma, micro- and macronutrient
values, etc [17]. They are also highly sensitive to active
technological impact factors. During protopectin
fragmentation, organic and mineral components can
enter into uncontrolled interactions, resulting in a partial
or complete loss of their biological potential. Therefore,

when assessing the native pectin potential, we should
take into account the presence of these biologically
active components among other significant factors.
Thus, we decided a complex operator as an
independent variable, taking into account mass fractions
of protopectin and biologically active components in the
tissue:
1
1
pp
i i pp
λ
ω
ϕ
ω ω
=
=
Σ + (2)
where ωpp is the mass fraction of protopectin, mg in
100 g; i ω is the mass fraction of the i-th biologically
active component, mg/100 g; and λ is the number of
biologically active components in the tissue (λ ∈ ¥N).
To apply this operator in practice, we transformed it
as follows:
1
1
1
1 1
1
1 i i
pp
λ ϕ
ω μ
ω
=
= =
+
+ Σ (3)
where
1
1
i i
pp
λ ω
μ
ω
= = Σ (4)
Thus 1
μ is the first dimensionless individual criterion
of protopectin fragmentation potential.
As we can see, with all possible values of ωpp
and i 1 i
λ ω
= Σ , this criterion has the following range of
definition:
μ1 ∈[0; ∞) (5)
In this case, Harrington’s individual desirability
function can be expressed as:
( )
10 1110 11 1 1 1
1
b b
d e e b b e e ϕ μ
 
− +  = − − + ⋅ = −  +  (6)
where d1 is the dependent dimensionless variable; b10
is the empirical dimensionless constant; and b11 is the
empirical dimensionless coefficient.
To determine the numerical values of b10 and b11,
we had to set the primary relations between the pairs
{μ11;d11} and {μ12 ;d12}, for which we proceeded from
the following considerations.
If an i-th biologically active component has a specific
measure of value pi, the total measure of value for all
biologically active components under consideration is:
bac i 1 i i 100 i 1 i i
v m p m p λ λ ω
= =
=Σ ⋅ = ⋅Σ ⋅ (7)
where vbac is the total measure of value for all
biologically active components, units; mi is the mass of
the i-th component, mg/100 g of plant tissue; m is the
tissue mass, mg; pi is the specific measure of value of
the i-th component, units/mg; and i ω is the mass fraction
of the i-th component in the plant tissue, %.
If specific measures of value for the components are
expressed through some average specific measure of
value
1
1
i i i
av
i i
m p
p
m
λ
λ
=
=
⋅
= Σ
Σ (8)
then formula (7) looks as follows:
100 1 100 1
av
bac i i av i i
m m p v p λ λ ω ω
= =
⋅
= ⋅Σ ⋅ = ⋅Σ (9)
from which
1
bac 100
i i
av
v
p m
λ ω
=
⋅
=
⋅ Σ (10)
If we apply similar considerations to protopectin,
then:
100
pp
pp pp pp pp
m
v m p p
⋅ω
= ⋅ = ⋅ (11)
where vpp is the total measure of protopectin value,
units; mpp if the mass of protopectin in the tissue,
mg; ppp is the specific measure of protopectin value,
units/mg; and ωpp is the mass fraction of the i-th
component in the plant tissue, %.
From Eq. (11), it follows that
pp 100
pp
pp
v
p m
ω
⋅
=
⋅ (12)
Thus, formula (4) can be presented as:
1
bac pp
pp av
v p
v p
μ
⋅
=
⋅ (13)
Grouping similar values on its sides, formula (13) can
be transformed as:
bac 1 av
pp pp
v p
v p
μ ⋅
= (14)
Respectively, if
bac 1
pp
v
v
>
, protopectin fragmentation
makes no sense, even with its significant amount in
the tissue. Therefore, a prerequisite for protopectin
fragmentation is:
1
pp
av
p
p
μ ≤ (15)
If pav is expressed as pav – in fractions of ppp, – then
condition (15) looks as follows:
1
μ1 pav ≤ − (16)
When calculating pav, it is advisable to use pi rather
than pi, its value reduced to ppp:
1
1
av i i i
av
pp i i
p m p
p
p m
λ
λ
=
=
⋅
= = Σ
Σ (17)
Theoretically, pi can be determined using several
approaches. However, we believe that the most
appropriate approach is based on a daily human need for
individual nutrients. This approach is least opportunistic
(compared to the financial approach) and subjective
(compared to direct expert assessments). Naturally, daily

Table 1 Specific measures of value for biologically active components and pectin in 100 g of plant tissue
Component Recommended daily
requirement, units
Estimated daily requirement Specific measure of value, mg-1
mg mg/kg pi pi
1 2 3 4 5 6
Protein, g 800.00III
Amino acids, mg/kgIII
– essential amino acids:
histidine 14 0.071428571 2.198
isoleucine 19 0.052631579 1.619
leucine 42 0.023809524 0.733
lysine 38 0.026315789 0.81
methionine 13.16I 0.075987842 2.338
phenylalanine + tyrosine 27 0.037037037 1.14
threonine 16 0.0625 1.923
tryptophan 4 0.25 7.692
valine 19 0.052631579 1.619
cysteine 5.84I 0.171232877 5.269
– non-essential amino acids 514.15II 0.001944958 0.06
– other amino acids 87.85IV 0.011383039 0.35
Lipids, gV 69.9 69 900 1 075.38
– saturated fatty acids 21.2 21 200 326.15 0.003066074 0.094
– monounsaturated fatty acids 25.4 25 400 390.77 0.00255905 0.079
– polyunsaturated fatty acids 23.3 23 300 358.46 0.002789712 0.086
Digestible carbohydrates, gVI 275 275 000 4 230.77 0.000236364 0.007
Pectin, gVII 2 2 000 30.77 0.0325 X 1
MineralsVIII
– Ca, mg 1 000 1 000 15.38462 0.064999981 2
– Mg, mg 400 400 6.15385 0.162499898 5
– K, mg 2 500 2 500 38.46154 0.025999999 0.8
– Na, mg 1 300 1 300 20 0.05 1.538
– P, mg 800 800 12.30769 0.081250015 2.5
– Cl, mg 2 300 2 300 35.38462 0.028260866 0.87
– Fe, mg 14.4 14.4 0.22154 4.513857543 138.888
– Zn, mg 12 12 0.18462 5.416531253 166.663
– J, μg 150 0.15 0.00231 432.9004329 13 320.013
– Cu, mg 1 1 0.01538 65.01950585 2 000.6
– Mn, mg 2 2 0.03077 32.49918752 999.975
– Se, μg 63 0.063 0.00097 1 030.927835 31 720.856
– Cr, μg 50 0.05 0.00077 1 298.701299 39 960.04
– Mo, μg 70 0.07 0.00108 925.9259259 28 490.028
– Co, μg 10 0.01 0.00015 6 666.666667 205 128.205
– Si, mg 30 30 0.46154 2.166659444 66.666
– F, mg 4 4 0.06154 16.24959376 499.988
Vitamins and provitamin IX
– water soluble
ascorbic acid (vitamin C), mg 90 90 1.38462 0.722219815 22.222
thiamine (vitamin B1), mg 1.5 1.5 0.02308 43.32755633 1 333.156
riboflavin (vitamin B2), mg 1.8 1.8 0.02769 36.11412062 1 111.204
vitamin B6, mg 2 2 0.03077 32.49918752 999.975
vitamin B12, μg 3 0.003 0.00005 20000 615 384.615
niacin, mg 20 20 0.30769 3.250024375 100.001
pantothenic acid, mg 5 5 0.07692 13.00052002 400.016
biotin, μg 50 0.05 0.00077 1298.701299 39 960.04
folic acid and folates, μg 400 0.4 0.00615 162.601626 5 003.127
– fat soluble
carotenoids, mg 5 5 0.07692 13.00052002 400.016
vitamin D, μg 10 0.01 0.00015 6 666.666667 205 128.205
352
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361
requirements for certain components depend on our
knowledge of biochemical processes in the human body,
as well as on the constantly changing environmental
situation in the world [24]. However, these factors should
not significantly affect pav.
The value of pav was calculated in several stages.
At the first stage, we determined daily requirements
for each of the biologically active components (ui) and
pectin (ups) based on a daily energy requirement of
2000 kcal and an average body weight of 65 kg. The
differences in daily requirements for men and women
were averaged. For comparability, all the values were
presented in mg/kg of body weight.
At the second stage, we calculated specific measures
of value for biologically active components ( pi) and
pectin ( pps):
1
pi ui= − (18)
1
pps ups = − (19)
The specific measures of value for pectin pps
and protopectin ppp were numerically identical since
protopectin is only valuable for the human body in the
form of its fragmentation products. To simplify, we
assumed that processing resulted in all protopectin
fragmented in a targeted manner (i.e., into fragments
that could be identified as pectin).
At the third stage, we determined specific measures
of value in the fractions of the specific measure of pectin
values pi.
The calculation results are shown in Table 1.
At the fourth stage, we calculated the value of 1
pav −
(Table 2). Based on the data in [31], we determined
the content of biologically active components in 100 g
of tissue for 21 types of plant materials from the
classification presented in [16]. For each type of raw
material, formula (17) was used to calculate the values of
pav ( j) and 1
pav ( j) − , where j ∈ ¥N.
Some assumptions were made in the calculations.
For example, the mass fractions of the components
which were not available in the database were assumed
as equal to zero [31]. The amount of carotenoids
was calculated based on the biological potential of
each type of raw material as 1 (o.c)
1
2
n
car car i i m mβ − = m
= + ⋅Σ ,
where mβ −car is the mass fraction of β-carotene,
mg/100 g; 1 (o.c)
n
i i m = Σ is the sum of mass fractions of
other carotenoids, mg/100 g [24]. The amount of
tocopherols was also calculated taking into account
the biological potential of each type of raw material
as 1
tok toc 10 toc m = mα − + ⋅mγ − , where mα −toc and mγ −toc are the
mass fractions of α- and γ-tocopherols, respectively;
mg/100 g [24]. To determine the sum of the remaining
amino acids, we subtracted the mass fractions of
essential and non-essential amino acids from the mass
fraction of protein.
The calculation results are shown in Table 2.
Since 1
pav ( j) − values were significantly different
for different types of raw materials, we calculated the
average 1
pav (av) − and the margin of error Δ to determine
boundary values (μ11 and μ12):
1
1 1 ( )
( )
j av j
av av
p
p
ζ
ζ
−
− = =
Σ (20)
( )
( )
( )
1 1 2
1 ( ) ( )
; 1 1
j av j av av p p
t
ζ
α ζ ζ ζ
− −
=
−
−
Δ = ⋅
⋅ −
Σ (21)
where ζ is the number of raw material types; t(α ;ζ −1) is
Student’s t-test; and α is the probability of error (0.05).
Based on the above, the value of μ11 for the first pair
{μ11;d11} was calculated as:
vitamin E, mg 15 15 0.23077 4.333318889 133.333
vitamin K, μg 120 0.12 0.00185 540.5405405 16 632.017
– pseudo-vitamins
inositol, mg 500 500 7.69231 0.129999961 4
L-carnitine, mg 300 300 4.61538 0.216666883 6.667
coenzyme Q10 (ubiquinone), mg 30 30 0.46154 2.166659444 66.666
lipoic acid, mg 30 30 0.46154 2.166659444 66.666
vitamin U, mg 20 20 0.30769 3.250024375 100.001
orotic acid (B13), mg 30 30 0.46154 2.166659444 66.666
paraminobenzoic acid, mg 100 100 1.53846 0.65000065 20
choline, mg 500 500 7.69231 0.129999961 4
Flavonoids, mgVIII 250 250 3.84615 0.26000026 8
I – according to [24] and the ratio in [25]
II – according to the ratio between essential and non-essential amino acids in [25]
III – according to the recommended dietary allowance in [24]
IV – the value is a difference between the daily requirement for protein and the sum of essential and non-essential amino acids
V – according to [24] and [26], based on a daily energy requirement of 2,000 kcal
VI – according to [27] and [28]
VII – according to [18]
VIII – according to [28]
IX – according to [28] and [29, 30]
X – the value corresponds to pps
1 2 3 4 5 6
Continuation of the table 1
353
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361
1
μ11 pav (av) = − − Δ (22)
The value of μ12 for the second pair {μ12 ;d12} was
calculated as the second order of μ11:
( )1 2
μ12 pav (av) = − − Δ (23)
The critical (boundary) values of 1
μ were based on
the analysis of Harrington’s desirability function, using
μ11 and μ12 as reference values. Since they are preset,
the calculated values were rounded to the nearest whole
number.
Despite the rigor of expression (16), its righthand
side is an empirical value based on the chemical
composition of a finite number of plant raw materials
and, therefore, it cannot be considered a priori. To make
up for this feature, we further determined the critical
values of 1 μ on the basis of Harrington’s desirability
function, using μ11 and μ12 as reference values.
Since a smaller reference value corresponded to
a larger value of Harrington’s individual desirability
function, we defined a condition Condd1 that determined
the individual form of the function as:
1
11 12
11 12
: 0.60 ; : 0.40
d 3 10
Cond d d
μ μ
 
=  ⇔ ⇔ 
 
(24)
Based on Condd1, we calculated the values of the
constant and the coefficient: b10 = −0.246; b11 = 3.673.
The critical values of the first criterion for the
protopectin fragmentation potential at the points with
standard critical values of the desirability function can
be calculated using Eq. (6) with the variable 1
μ:
( )
11
1
10 1
[ ] 1
ln ln i
i
D b
b d
μ = − −
+ − 
(25)
where 1[ ] i D μ is the value of the criterion 1
μ at the critical
Table 2 Weighted average reduced measures of raw materials value in non-uronide biologically active components
Raw materials pav ( j) 1
pav ( j) − Raw materials ( ) pav j 1
pav ( j) −
Carrot 0.8282 1.207 Persimmon 0.0887 11.274
Beetroot 0.4156 2.406 Grapefruit 0.2355 4.246
Watermelon 0.2860 3.497 Lemon with skin 0.6618 1,511
Pumpkin 0.6578 1.520 without skin 0.3057 3,271
Melon 0.1749 5.718 Orange 0.2729 3.664
Apples 0.0783 12.771 Tangerine 0.1691 5.914
Quince 0.0993 10.070 Currants red 0.2654 3,768
Pears 0.0889 11.249 black 0.4389 2,278
Figs 0.1137 8.795 Cranberry 0.3147 3.178
Pomegranate 0.1671 5.984 Gooseberry 0.2935 3.407
Grapes 0.1057 9.461 Feijoa 0.2074 4.822
Figure 1 Graphic interpretation of Harrington’s individual desirability function given condition 1 d Cond and variable 1
μ
D1 D2 D3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
d1
μ1
I II III IV
D1 D2 D3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
d1
μ1
I II III IV
354
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361
point Di of Harrington’s individual desirability function
determined by Eq. (6) and corresponding to d1i; and
d1i is the standard i-th critical (canonical) value d1 of
Harrington’s individual desirability function.
The graphic interpretation of Harrington’s individual
desirability function corresponding to the condition
Condd1 is given in Fig. 1. For each value of d1i, we
determined the corresponding values of μ1[Di ].
As we can see, the 1
μ range of definition includes
four domains separated by the critical values of μ1[Di ],
where i = 1, 2, 3. By definition, domain IV includes those
1 μ
values at which the fragmentation of the protopectin
complex makes no sense due to a low value of the
individual function of desirability.
Domain III covers those 1
μ values at which the
individual desirability function is large enough
for protopectin fragmentation to make sense, but
insufficiently large to neglect non-uronide bioactive
components and isolate the products of fragmentation.
In domains I and II, the individual desirability
function is so large that the content of non-uronide
bioactive components in plant tissue can be completely
ignored.
Based on the physical meaning of the boundary
conditions for 1
μ, we established two individual
boundary conditions that partially determined the native
pectin potential of plant tissue.
Boundary condition I:
– μ1 > μ1[D3 ] means the absence of the first individual
potential for protopectin fragmentation;
– μ1 ≤ μ1[D3 ] means the presence of the first individual
potential for protopectin fragmentation.
Boundary condition II:
– μ1[D3 ] ≥ μ1 > μ1[D2 ] means the absence of the first
individual potential for isolation of protopectin
fragmentation products;
– μ1 ≤ μ1[D2 ] means the presence of the first individual
potential for isolation of protopectin fragmentation
products.
Next, we determined the structure and properties
of the second dimensionless individual criterion for the
protopectin fragmentation potential.
The second independent variable was a complex
operator based on the mass fraction of protopectin in the
tissue:
2 100 2
ωpp
ϕ = = μ (26)
where ϕ2 is the dimensionless operator of Harrington’s
individual desirability function; and μ2 is the second
dimensionless individual criterion for the protopectin
fragmentation potential.
Harrington’s individual desirability function was
expressed as:
( 20 21 2) ( 20 21 2)
2
d e e b b e e b b = − − + ⋅ϕ = − − + ⋅μ (27)
Thus, the condition Condd2 that determined the
individual function was set as:
2
21 22
21 22
: 0.35 ; : 0.65
d 0.001 0.05
Cond d d
μ μ
 
=  ⇔ ⇔ 
 
(28)
Based on Condd2, we calculated the values of the
constant and the coefficient: 2
b20 6.68 10= − ⋅ − and
b21 = 18.179. The critical values of the μ2 criterion were
calculated as:
( ) 20 2
2
21
ln ln
[ ] i
i
b d
D
b
μ
+ −  = − (29)
where μ2[Di ] is the value of μ2 at the critical point
Di of Harrington’s individual desirability function
calculated by Eq. (6) and corresponding to d2i; d2i is the
standard i-th critical (canonical) value d2 of Harrington’s
individual desirability function.
Figure 2 Graphic interpretation of Harrington’s individual desirability function given condition Condd2 and variable 2 μ
D3 D2 D1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
d2
μ2
IV III II I
D1 D2 D3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
d1
μ1
I II III IV
×
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Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361
The graphic interpretation of Harrington’s individual
desirability function corresponding to the condition
Condd2 is presented in Fig. 2. For each value of 2i d , we
calculated the corresponding values of μ2[Di ].
Just like with 1 μ , the μ2 range of definition includes
four domains separated by the critical values of μ2[Di ],
where i = 1, 2, 3.
By definition, domain IV covers those values of μ2
at which the fragmentation of the protopectin complex
makes no sense. This led us to formulate the third
individual boundary condition:
– μ2 < μ2[D3 ] means the absence of the second
individual potential for protopectin fragmentation;
– μ2 ≥ μ2[D3 ] means the presence of the second
individual potential for protopectin fragmentation.
We should note that fragmentation potentials I and
II are categorical, i.e., if one of them is absent, the total
fragmentation potential is absent as well.
Domains I, II, and III include such values of μ2 that
ensure not only protopectin fragmentation, but also
the isolation of fragmentation products. Based on the
canonical reference values of the individual desirability
function, we formulated the fourth boundary condition:
– μ2[D3 ] ≤ μ2 < μ2[D1] means the absence of the
second individual potential for isolation of protopectin
fragmentation products;
– μ2 ≥ μ2[D1] means the presence of the second
individual potential for isolation of protopectin
fragmentation products.
Similar to the first and the second fragmentation
potentials, the individual isolation potentials are
categorical.
The third independent variable was a complex
operator based on the mass fraction of pectin substances
in the tissue:
3 100 3
ωps
ϕ = = μ (30)
where ϕ3 is the dimensionless operator of Harrington’s
individual desirability function; ωps is the total
amount of pectin substances, %; and μ3 is the third
dimensionless individual criterion for the protopectin
fragmentation potential.
In this case, the condition Condd3 that determined the
individual function was calculated as:
3
31 32
31 32
: 0.40 ; : 0.65
d 0.01 0.07
d d
Cond
μ μ
 
=  ⇔ ⇔ 
 
(31)
Based on expression (31), we calculated the
constant and the coefficient as 2
b30 3.8367 10= − ⋅ − and
b31 = 12.5788, respectively, and the critical boundaries of
μ3, as:
( ) 30 3
3
31
ln ln
[ ] i
i
b d
D
b
μ
+ −  = − (32)
where μ3[Di ] is the value of μ3 at the critical point
Di of Harrington’s individual desirability function
calculated by (6) and corresponding to d3i; and d3i is the
standard i-th critical (canonical) value d3 of Harrington’s
individual desirability function.
Figure 3 shows the graphic interpretation of
Harrington’s individual desirability function given
Condd3. For each value of d3i, we calculated the
corresponding values of μ3[Di ].
Here, we can clearly see domain IV with no pectin
potential in the plant tissue.
As a result, we formulated the fifth individual
boundary condition:
– μ3 < μ3[D3 ] means the absence of pectin potential;
– μ3 ≥ μ3[D3 ] means the presence of pectin potential.
Thus, the pectin potential is categorical.
The fourth independent variable was a complex
operator based on the ratio of the mass fractions of
protopectin and pectin substances in the tissue:
4
4
1
1
pp
ps
ω
ϕ
ω μ
= =
+ (33)
where ϕ4 is the dimensionless operator of Harrington’s
individual desirability function; ωsp is the mass fraction
of natively soluble pectin substances, %; and μ4 is
the third dimensionless individual criterion for the
protopectin fragmentation potential calculated as:
4
sp
pp
ω
μ
ω
= (34)
Then, the condition
Condd4, which determined the
individual function, was calculated as:
4
41 42
41 42
: 0.65 ; : 0.80
d 2.50 1.25
Cond d d
μ μ
 
=  ⇔ ⇔ 
 
(35)
Based on expression (35), we calculated the constant
and the coefficient (b40 = −0.3419, b41 = 4.1441).
Based on
Condd4, the critical boundaries of μ4 were
calculated as:
( )
41
4
40 4
[ ] 1
ln ln i
i
D b
b d
μ = − −
+ − 
(36)
where μ4[Di ] is the value of μ4 at the critical point
Di of Harrington’s individual desirability function
calculated by (6) and corresponding to d4i; and d4i is the
standard i-th critical (canonical) value d4 of Harrington’s
individual desirability function.
Figure 4 shows the graphic interpretation of
Harrington’s individual desirability function given
Condd4, with d4i values corresponding to μ4[Di ] values.
Based on the logical content of d4i and the numerical
values of μ4[Di ], the range of definition can be divided
into four domains that determine the fragmentation
potential of the protopectin complex and the isolation
potential of fragmentation products.
According to Fig. 4, domain IV covers those values
μ4 at which the mass fraction of water-soluble pectin
exceeds that of the protopectin complex so much
that there is practically no reason for its individual
fragmentation. Thus, we determined the sixth boundary
condition as follows:
– μ4 > μ4[D3 ] means the absence of the third individual
potential for protopectin fragmentation;
– μ4 ≤ μ4[D3 ] means the presence of the third
individual potential for protopectin fragmentation.
×
356
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361
Figure 3 Graphic interpretation of Harrington’s individual desirability function given condition Condd3 and variable μ3
Figure 4 Graphic interpretation of Harrington’s individual desirability function given condition
Condd4 and variable 4 μ D1 D2 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 d1
I II III D1 D2 D3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
d4
μ4
I II III IV
D1 D2 D3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
d1
μ1
I II III IV
D3 D2 D1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30
d3
μ3
IV III II I
Following the same pattern, we determined the seventh
boundary condition (VII), namely:
– μ4[D2 ] < μ4 ≤ μ4[D3 ] means the absence of the
third individual potential for isolation of protopectin
fragmentation products;
–μ4 ≤ μ4[D1] means the presence of the third individual
potential for isolation of protopectin fragmentation
products.
In addition, boundary conditions VI and VII are
based on:
μ4 ≤ μ4[Di ] (37)
where i = 3 for condition VI and, i = 2 for condition VII.
However, μ4 can be expressed as:
3 2
4
2
sp ps pp
pp pp
ω ω ω μ μ
μ
ω ω μ
− −
= = = (38)
Then, given the presence of the third individual
fragmentation potential:
μ3 ≤ μ2 ⋅ (μ4[Di ]+1) (39)
Thus, the third individual potentials of fragmentation
and isolation are relative since they are involved in
the formation of respective total potentials indirectly,
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Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361
through expressions in which they act as one of the
variables.
If we assume that there is a certain criterion space
with coordinates μ1, μ2 and μ3, the pectin potential
of any plant material can be clearly determined as
a geometrical location of the point M[μ1, μ2 , μ3 ]
corresponding to the material under analysis.
Based on the a priori assumption that
1 ps i i 100
λ ω ω
=
+Σ ≤ (40)
we can establish the eighth boundary condition (VIII):
the top boundary of the range of definition for all
possible values of M[μ1, μ2 , μ3 ] is determined by the
following basic proposition:
μ3(top) = 1−μ1 ⋅μ2 (41)
In addition, since a part cannot be larger than a
whole, it is also true that:
ωpp ≤ωps (42)
which leads to the following condition:
μ3 ≥ μ2 (43)
i.e., the bottom boundary of the range of definition for
all possible values of M[μ1, μ2 , μ3 ] is determined by the
second basic proposition:
μ3(bot) = μ2 (44)
The last formula is an expression of boundary
condition IX.
RESULTS AND DISCUSSION
Thus, according to boundary conditions VIII and
IX, a set (A) of all points M[μ1, μ2 , μ3 ] can be defined as
M[ 1, 2 , 3 ] [ 3(bot) , 3(top) ] μ1 0;μ2 0;μ3 0 μ μ μ ∈ μ μ ≥ ≥ ≥ , (45)
graphically presented in Fig. 5.
The logic of assessing plant bioresources for the
presence of pectin substances determines general
boundary conditions for defining a set of points
M[μ1, μ2 , μ3 ] as the following hierarchy: “individual
pectin potential → individual fragmentation potential of
the protopectin complex → individual isolation potential
of protopectin fragmentation products”. Thus, the entire
set of points M[μ1, μ2 , μ3 ] can be divided into four
subsets:
– subset A1 characterized by the absence of a common
pectin potential in all the elements;
– subset A2 where A2 ∩ A1 = ∅ and all the elements have
a common pectin potential, but lack a common potential
for protopectin fragmentation;
– subset A3 where A3 ∩ A2 = ∅ and all the elements
have common pectin and protopectin fragmentation
potentials, but lack a common isolation potential for
fragmentation products; and
– subset A4 where A4 ∩ A3 = ∅ and all elements
have common pectin and protopectin fragmentation
potentials, as well as isolation potential for
fragmentation products.
By definition, the following is true for all the subsets:
A1 ∩ A2 ∩ A3 ∩ A4 = ∅ (46)
Based on the above, the existence of A1 corresponds
to:
μ2 ≤ μ3 < μ3[D3 ] (47)
The area of definition for all A1 elements is partially
presented in Fig. 6.
The existence of subset A2 correspond2s to: 4 3 2 2 3
3 3 2 2 3
1 2 3
2 2 3 3
1 1 3
3 3 2 3 3
( [ ] 1), [ ]
[ ], [ ]
1
, [ ]
[ [ ], [ ]
D D
D D
D
D
D D
μ μ μ μ
μ μ μ μ
μ μ μ
μ μ μ
μ μ
μ μ
 ⋅ + ≥   < − ⋅ ≥ ≥
 ≥  >  <
2 4 3 2 2 3
1 1 3
3 3 2 2 3
1 2 3
2 2 3 3
1 1 3
3 3 2 3 3
( [ ] 1), [ ]
[ ]
[ ], [ ]
1
, [ ]
[ ]
[ ], [ ]
D D
D
D D
D
D
D D
μ μ μ μ
μ μ
μ μ μ
μ μ μ
μ μ μ
μ μ
μ μ μ
 ⋅ + ≥  ≤  < − ⋅ ≥ ≥
 ≥  >  <
(48)
Figure 7 shows a partial area of definition for all A2
elements.
The existence of A3 corresponds to: 1 2 4 2
1 2 1 1 3 2 3
2 4 3 2 2 3 1 3 1 1 2
2 3 2 2 ( [ ] 1)
1 [ ] ( [ ] 1) [ ] [ ] [ [ ] [ D
D D D D D
D μ μ μ
μ μ μ μ μ μ
μ μ μ μ μ μ μ
μ
μ μ μ
 ≤
 ⋅ +  − ⋅ ≤  ≥  > ≥  ⋅ + ≥ ≥ >  

1 1 2  ≤ <
2 4 2
1 2 1 1 3 2 2 1
3
2 4 3 2 2 3 1 3 1 1 2
2
2 3 2 2 1
[ ]
( [ ] 1)
1 [ ] [ ]
( [ ] 1) [ ] [ ] [ ]
[ ] [ ]
D
D
D D
D D D D
D D
μ μ
μ μ
μ μ μ μ μ μ
μ
μ μ μ μ μ μ μ
μ
μ μ μ
 ≤
 ⋅ +  − ⋅ ≤  ≥  > ≥  ⋅ + ≥ ≥ >  

 ≤ <
(49)
Figure 8 presents the area of definition for all A3
elements.
The existence of subset A4 corresponds to:
1 2 1 1 2
3 2
2 4 2 2 2 1
1 [ ]
( [ ] 1) [ ]
D
D D
μ μ μ μ
μ μ
μ μ μ μ
 − ⋅  ≤  > ≥   ⋅ +  ≥
(50)
The area of definition for all A4 elements is presented
in Fig. 9.
Thus, the specific value M[μ1, μ2 , μ3 ] that shows its
belonging to one of the subsets Ai (where i = 1, 2,3, 4) in
the zoned criterion space clearly determines the plant
tissue’s overall potential for protopectin fragmentation.
Our approach to classifying plants as pectincontaining
materials, which is based on a system of
criteria and a zoned criterion space, has clear advantages
over existing methods due to its objectivity determined
by the boundary conditions.
However, when analyzing this approach, we can
easily see that the μ j1 and μ j2 values corresponding
to d j1 and d j2 in the conditions Condd j j=2,3,4 were set
a priori, based on general assumptions regarding the
degree of acceptability of certain μ j values within
Harrington’s individual desirability functions in
accordance with the boundary (canonical) values of d.
Yet, the conditions Condd j j=2,3,4 determine the coefficients
and constants, and, consequently, individual desirability
358
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361
functions, as well as numerical values of μ j [Di ].
Therefore, at this stage, our approach has a general,
conceptual form requiring further research.
Based on the results, we developed a generalized
algorithm to determine the geometric location of plant
tissue in the zoned criterion space, or M[μ1, μ2 , μ3 ]
belonging to one of the subsets (Fig. 10). We can use
this algorithm to assess any plant tissue’s potential
for transformation of the protopectin complex, which
determines the influence of any technological impact on
its pectin potential.
The approach that we used to determine the criterion
space and boundary conditions for its zoning explicitly
suggests that this algorithm is universal for classifying
plant tissue or its derivatives as pectin-containing
materials. Thus, the algorithm is applicable to any type
of plant material for which the μ1, μ2 and μ3 criteria can
be numerically expressed.
CONCLUSION
To sum up, our investigation showed the following
results.
1. We developed a system of criteria to assess the
transformation potential of the protopectin complex in
plant tissue. This system is based on the geometrical
Figure 6 Partial definition area for subset А1 Figure 5 Definition area of the criterion space
Figure 7 Partial definition area for subset A2 Figure 8 Partial definition area for subset A3
Figure 9 Partial definition area for subset A4
359
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361
location of M[μ1, μ2 , μ3 ] – the point that corresponds
to the material under analysis – in a zoned criterion
space with coordinates in the form of dimensionless
individual criteria for protopectin fragmentation
potential.
2. The dimensionless individual criteria for
protopectin fragmentation potential included the ratio
between the mass fractions of biologically active
components and protopectin in plant tissue, the mass
fraction of the protopectin complex expressed in
unit fractions, and the mass fraction of total pectin
substances expressed in unit fractions.
3. We established nine individual boundary
conditions, individual pectin potential, two individual
fragmentation potentials, and three individual isolation
potentials for pectin substances, which altogether
determine a system of zoning the criterion space.
4. The boundary conditions in the definition area
for a set of points M[μ1, μ2 , μ3 ] had the following
hierarchy: individual pectin potential → individual
Start
𝜔𝑠𝑝; 𝜔𝑝𝑝; 􀷍 𝜔𝑖
𝜆
𝑖=1
𝜇1; 𝜇2; 𝜇3; 𝜇4
𝜇1[𝐷𝑖]|𝑖=2,3; 𝜇2[𝐷𝑖]|𝑖=1,3; 𝜇3[𝐷𝑖]|𝑖=3; 𝜇4[𝐷𝑖]|𝑖=2,3
𝜇3 < 𝜇3[𝐷3]
𝜇1 > 𝜇1[𝐷3]
𝜇2 < 𝜇2[𝐷3]
𝜇3 ≥ 𝜇2 ∙ (𝜇4[𝐷3] + 1)
𝜇2 < 𝜇2[𝐷1]
𝜇1 > 𝜇1[𝐷2]
𝜇3 ≥ 𝜇2 ∙ (𝜇4[𝐷2] + 1)
Finish
protopectin complex’
fragmentation with
product isolation
without protopectin
complex’ fragmentation
protopectin complex’
fragmentation without
product isolation
pectic
potential is absent
yes no
no no
no yes
yes
no
yes no
yes no
yes
no
А1
А2
А3 А4
Figure 10 Algorithm for plant tissue classification according to protopectin fragmentation potential based on the geometric
location in the zoned criterion space
360
Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361
fragmentation potential of the protopectin complex
→ individual isolation potential of protopectin
fragmentation products.
5. We developed an algorithm to classify plant
tissues according to protopectin fragmentation potential
based on the geometric location in the zoned criterion
space.
CONTRIBUTION
All the authors were equally involved in writing the
manuscript and are equally responsible for plagiarism.
CONFLICT OF INTEREST
The authors state that there is no conflict
of interest.

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