INTRODUCTIONFood technology is currently striving to maximizethe potential of raw materials and use new, nontraditionalsources of essential nutraceuticals andfood components with biological (antioxidants,enterosorbents, etc.) and/or technological (thickeners,stabilizers, etc.) functional activity [1, 2]. The mostpromising way to achieve that is a biotechnologicalapproach that makes use of both living cultures ofmicroorganisms and isolated enzyme systems. Whenusing isolated enzyme systems, this approach involves amultiple stage fragmentation of a native supramolecularcomplex of plant and/or animal cell walls into targetcomponents with a wide range of physicochemical and/or technological properties [3–5].One of the methods within this approach is toactivate the potential of a multicomponent polymermatrix of cell walls and intercellular spaces. Thismethod has a limited use in processing agriculturalraw materials. It mainly consists in partial or completedegradation (depolymerization) of its individualcomponents to change the consistency or transparencyof the final product, or to clear it of degradation productsand improve its sensory characteristics. Most certainly,a targeted use of this polymer matrix is complicated byits highly heterogeneous components, a system of bondsbetween them, and highly entangled polymer chains [6].Moreover, the heterogeneity of individual matrixcomponents is a serious obstacle to controlling theirproperties during extraction [7, 8].Pectin substances are among major carbohydratebiopolymers that have a wide variety of functional andtechnological characteristics [9, 10]. In a plant cell,they are represented by two main fractions – nativewater-soluble pectin and a native water-insolubleprotopectin complex. The last one is the most valuablefor transformation due to its molecular structure andcomposition [9].The structure of cell walls in almost all terrestrialplants [6, 11, 12] makes them a potentially good resourcefor the industrial production of pectin. However, it isdifficult to implement. Since the protopectin complex isa branched supramolecular structure incorporated intothe cell wall, its transformation is mainly fragmentationinto water-soluble polymers (soluble pectin). In addition,mass fractions of pectin substances and the protopectincomplex may depend on the type, grade, and purpose ofraw materials, their structure and phase of development,soil and weather conditions for their vegetation, aswell as localization, duration and storage conditions,processing intensity, etc. [10, 13]. In this regard, thechoice of a plant as a pectin-containing material shouldbe determined by the purpose of its use.Raw materials can be classified according to thesize of their pectin potential – “high”, “medium,” and“small” (“low”, “insignificant”) [9, 10, 14]. The onlyfundamental approach to pectin production was offeredby Donchenko in [15] and supplemented by Rodionovaet al. in [19, 20] (works [16–18] are actualy based on[15]). Although this approach is rather fragmented, itcan be used as a basis for developing a universal systemthat takes into account the native pectin potentialof plant tissue.The protopectin complex is a key object whosefragmentation enables us to use the biomass of a plantmaterial as a source of pectin substances. Due to thepresence of certain plant organisms, mainly a nativelysoluble fraction of pectin, biomass can be attributedto potential sources of pectin. On the other hand, thebiomass of certain taxonomic elements may contain asmall amount of pectin, which makes its use ineffective.Therefore, we found it relevant to develop a clearcutclassification of plant bio-resources into groups todetermine the prospects of their use as pectin-containingraw materials.In this regard, we aimed to develop a system ofcriteria for assessing the transformation potential ofnative complexes of plant carbohydrate biopolymersexemplified by pectin. To achieve this aim, we set thefollowing objectives:– working out criteria to assess the transformationpotential of native plant biopolymers and the concept oftheir applicability, and– developing a system of boundary conditions andan universal algorithm for classifying plant materialsaccording to the transformation potential of their nativepectin components.STUDY OBJECTS AND METHODSAccording to existing data, all plant materials can beclassified into four main groups, namely:– bio-resources with sufficient potential for protopectinfragmentation and subsequent isolation of its products asindependent substances;– bio-resources with sufficient potential for protopectinfragmentation, but with insufficient potential forisolation of its products;– bio-resources with insufficient potential forprotopectin fragmentation, but with sufficient potentialfor natively soluble pectin;– bio-resources with no pectin potential.On the one hand, this differentiation involvesunifying plant characteristics and reducing them tocertain generalized values. On the other hand, it involvesdividing the domain of generalized values into four fixedzones. As we know, a universal tool for unifying anarbitrary set of source factors is a range of anonymizedcriteria reducible to a certain system with the use ofboundary conditions [21, 22]. Thus, we can apply acriteria-based approach to fulfilling our objectives.To be able to scale the criteria to determine clearboundary conditions, we used Harrington’s individualdesirability function in its canonical form [23]:e (bi0 bi1 i )di e = − − + ⋅ϕ (1)where di is the dimensionless value of Harrington’sindividual desirability function; bi0 is the constant; bi1 isthe coefficient; and φi is the dimensionless operator ofHarrington’s individual desirability function.We introduced the first and second individual criteriafor protopectin fragmentation potential among the maincriteria 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 certainamount of protopectin in the cell wall matrix is notsufficient for assessing the native pectin potential ofplant tissue. The tissues of many plant organisms alsocontain a significant amount of organic and mineralcomponents with valuable vitamins and antioxidantactivity, pronounced aroma, micro- and macronutrientvalues, etc [17]. They are also highly sensitive to activetechnological impact factors. During protopectinfragmentation, organic and mineral components canenter into uncontrolled interactions, resulting in a partialor complete loss of their biological potential. Therefore,when assessing the native pectin potential, we shouldtake into account the presence of these biologicallyactive components among other significant factors.Thus, we decided a complex operator as anindependent variable, taking into account mass fractionsof protopectin and biologically active components in thetissue:11ppi i ppλωϕω ω==Σ + (2)where ωpp is the mass fraction of protopectin, mg in100 g; i ω is the mass fraction of the i-th biologicallyactive component, mg/100 g; and λ is the number ofbiologically active components in the tissue (λ ∈ ¥N).To apply this operator in practice, we transformed itas follows:1111 111 i ippλ ϕω μω== =++ Σ (3)where11i ippλ ωμω= = Σ (4)Thus 1μ is the first dimensionless individual criterionof protopectin fragmentation potential.As we can see, with all possible values of ωppand i 1 iλ ω= Σ , this criterion has the following range ofdefinition:μ1 ∈[0; ∞) (5)In this case, Harrington’s individual desirabilityfunction can be expressed as:( )10 1110 11 1 1 11b bd e e b b e e ϕ μ − +  = − − + ⋅ = −  +  (6)where d1 is the dependent dimensionless variable; b10is the empirical dimensionless constant; and b11 is theempirical 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 fromthe following considerations.If an i-th biologically active component has a specificmeasure of value pi, the total measure of value for allbiologically active components under consideration is:bac i 1 i i 100 i 1 i iv m p m p λ λ ω= ==Σ ⋅ = ⋅Σ ⋅ (7)where vbac is the total measure of value for allbiologically active components, units; mi is the mass ofthe i-th component, mg/100 g of plant tissue; m is thetissue mass, mg; pi is the specific measure of value ofthe i-th component, units/mg; and i ω is the mass fractionof the i-th component in the plant tissue, %.If specific measures of value for the components areexpressed through some average specific measure ofvalue11i i iavi im ppmλλ==⋅= ΣΣ (8)then formula (7) looks as follows:100 1 100 1avbac i i av i im m p v p λ λ ω ω= =⋅= ⋅Σ ⋅ = ⋅Σ (9)from which1bac 100i iavvp mλ ω=⋅=⋅ Σ (10)If we apply similar considerations to protopectin,then:100pppp pp pp ppmv 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-thcomponent in the plant tissue, %.From Eq. (11), it follows thatpp 100ppppvp mω⋅=⋅ (12)Thus, formula (4) can be presented as:1bac pppp avv pv pμ⋅=⋅ (13)Grouping similar values on its sides, formula (13) canbe transformed as:bac 1 avpp ppv pv pμ ⋅= (14)Respectively, ifbac 1ppvv>, protopectin fragmentationmakes no sense, even with its significant amount inthe tissue. Therefore, a prerequisite for protopectinfragmentation is:1ppavppμ ≤ (15)If pav is expressed as pav – in fractions of ppp, – thencondition (15) looks as follows:1μ1 pav ≤ − (16)When calculating pav, it is advisable to use pi ratherthan pi, its value reduced to ppp:11av i i iavpp i ip m ppp mλλ==⋅= = ΣΣ (17)Theoretically, pi can be determined using severalapproaches. However, we believe that the mostappropriate approach is based on a daily human need forindividual nutrients. This approach is least opportunistic(compared to the financial approach) and subjective(compared to direct expert assessments). Naturally, dailyTable 1 Specific measures of value for biologically active components and pectin in 100 g of plant tissueComponent Recommended dailyrequirement, unitsEstimated daily requirement Specific measure of value, mg-1mg mg/kg pi pi1 2 3 4 5 6Protein, g 800.00IIIAmino acids, mg/kgIII– essential amino acids:histidine 14 0.071428571 2.198isoleucine 19 0.052631579 1.619leucine 42 0.023809524 0.733lysine 38 0.026315789 0.81methionine 13.16I 0.075987842 2.338phenylalanine + tyrosine 27 0.037037037 1.14threonine 16 0.0625 1.923tryptophan 4 0.25 7.692valine 19 0.052631579 1.619cysteine 5.84I 0.171232877 5.269– non-essential amino acids 514.15II 0.001944958 0.06– other amino acids 87.85IV 0.011383039 0.35Lipids, 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 0.086Digestible carbohydrates, gVI 275 275 000 4 230.77 0.000236364 0.007Pectin, gVII 2 2 000 30.77 0.0325 X 1MineralsVIII– 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 499.988Vitamins and provitamin IX– water solubleascorbic acid (vitamin C), mg 90 90 1.38462 0.722219815 22.222thiamine (vitamin B1), mg 1.5 1.5 0.02308 43.32755633 1 333.156riboflavin (vitamin B2), mg 1.8 1.8 0.02769 36.11412062 1 111.204vitamin B6, mg 2 2 0.03077 32.49918752 999.975vitamin B12, μg 3 0.003 0.00005 20000 615 384.615niacin, mg 20 20 0.30769 3.250024375 100.001pantothenic acid, mg 5 5 0.07692 13.00052002 400.016biotin, μg 50 0.05 0.00077 1298.701299 39 960.04folic acid and folates, μg 400 0.4 0.00615 162.601626 5 003.127– fat solublecarotenoids, mg 5 5 0.07692 13.00052002 400.016vitamin D, μg 10 0.01 0.00015 6 666.666667 205 128.205352Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361requirements for certain components depend on ourknowledge of biochemical processes in the human body,as well as on the constantly changing environmentalsituation in the world [24]. However, these factors shouldnot significantly affect pav.The value of pav was calculated in several stages.At the first stage, we determined daily requirementsfor each of the biologically active components (ui) andpectin (ups) based on a daily energy requirement of2000 kcal and an average body weight of 65 kg. Thedifferences in daily requirements for men and womenwere averaged. For comparability, all the values werepresented in mg/kg of body weight.At the second stage, we calculated specific measuresof value for biologically active components ( pi) andpectin ( pps):1pi ui= − (18)1pps ups = − (19)The specific measures of value for pectin ppsand protopectin ppp were numerically identical sinceprotopectin is only valuable for the human body in theform of its fragmentation products. To simplify, weassumed that processing resulted in all protopectinfragmented in a targeted manner (i.e., into fragmentsthat could be identified as pectin).At the third stage, we determined specific measuresof value in the fractions of the specific measure of pectinvalues pi.The calculation results are shown in Table 1.At the fourth stage, we calculated the value of 1pav −(Table 2). Based on the data in [31], we determinedthe content of biologically active components in 100 gof tissue for 21 types of plant materials from theclassification presented in [16]. For each type of rawmaterial, formula (17) was used to calculate the values ofpav ( j) and 1pav ( j) − , where j ∈ ¥N.Some assumptions were made in the calculations.For example, the mass fractions of the componentswhich were not available in the database were assumedas equal to zero [31]. The amount of carotenoidswas calculated based on the biological potential ofeach type of raw material as 1 (o.c)12ncar car i i m mβ − = m= + ⋅Σ ,where mβ −car is the mass fraction of β-carotene,mg/100 g; 1 (o.c)ni i m = Σ is the sum of mass fractions ofother carotenoids, mg/100 g [24]. The amount oftocopherols was also calculated taking into accountthe biological potential of each type of raw materialas 1tok toc 10 toc m = mα − + ⋅mγ − , where mα −toc and mγ −toc are themass fractions of α- and γ-tocopherols, respectively;mg/100 g [24]. To determine the sum of the remainingamino acids, we subtracted the mass fractions ofessential and non-essential amino acids from the massfraction of protein.The calculation results are shown in Table 2.Since 1pav ( j) − values were significantly differentfor different types of raw materials, we calculated theaverage 1pav (av) − and the margin of error Δ to determineboundary values (μ11 and μ12):11 1 ( )( )j av jav avppζζ−− = =Σ (20)( )( )( )1 1 21 ( ) ( ); 1 1j av j av av p ptζα ζ ζ ζ− −=−−Δ = ⋅⋅ −Σ (21)where ζ is the number of raw material types; t(α ;ζ −1) isStudent’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 133.333vitamin K, μg 120 0.12 0.00185 540.5405405 16 632.017– pseudo-vitaminsinositol, mg 500 500 7.69231 0.129999961 4L-carnitine, mg 300 300 4.61538 0.216666883 6.667coenzyme Q10 (ubiquinone), mg 30 30 0.46154 2.166659444 66.666lipoic acid, mg 30 30 0.46154 2.166659444 66.666vitamin U, mg 20 20 0.30769 3.250024375 100.001orotic acid (B13), mg 30 30 0.46154 2.166659444 66.666paraminobenzoic acid, mg 100 100 1.53846 0.65000065 20choline, mg 500 500 7.69231 0.129999961 4Flavonoids, mgVIII 250 250 3.84615 0.26000026 8I – 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 acidsV – according to [24] and [26], based on a daily energy requirement of 2,000 kcalVI – 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 pps1 2 3 4 5 6Continuation of the table 1353Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–3611μ11 pav (av) = − − Δ (22)The value of μ12 for the second pair {μ12 ;d12} wascalculated as the second order of μ11:( )1 2μ12 pav (av) = − − Δ (23)The critical (boundary) values of 1μ were based onthe analysis of Harrington’s desirability function, usingμ11 and μ12 as reference values. Since they are preset,the calculated values were rounded to the nearest wholenumber.Despite the rigor of expression (16), its righthandside is an empirical value based on the chemicalcomposition of a finite number of plant raw materialsand, therefore, it cannot be considered a priori. To makeup for this feature, we further determined the criticalvalues of 1 μ on the basis of Harrington’s desirabilityfunction, using μ11 and μ12 as reference values.Since a smaller reference value corresponded toa larger value of Harrington’s individual desirabilityfunction, we defined a condition Condd1 that determinedthe individual form of the function as:111 1211 12: 0.60 ; : 0.40d 3 10Cond d dμ μ =  ⇔ ⇔  (24)Based on Condd1, we calculated the values of theconstant and the coefficient: b10 = −0.246; b11 = 3.673.The critical values of the first criterion for theprotopectin fragmentation potential at the points withstandard critical values of the desirability function canbe calculated using Eq. (6) with the variable 1μ:( )11110 1[ ] 1ln ln iiD bb dμ = − −+ − (25)where 1[ ] i D μ is the value of the criterion 1μ at the criticalTable 2 Weighted average reduced measures of raw materials value in non-uronide biologically active componentsRaw materials pav ( j) 1pav ( j) − Raw materials ( ) pav j 1pav ( j) −Carrot 0.8282 1.207 Persimmon 0.0887 11.274Beetroot 0.4156 2.406 Grapefruit 0.2355 4.246Watermelon 0.2860 3.497 Lemon with skin 0.6618 1,511Pumpkin 0.6578 1.520 without skin 0.3057 3,271Melon 0.1749 5.718 Orange 0.2729 3.664Apples 0.0783 12.771 Tangerine 0.1691 5.914Quince 0.0993 10.070 Currants red 0.2654 3,768Pears 0.0889 11.249 black 0.4389 2,278Figs 0.1137 8.795 Cranberry 0.3147 3.178Pomegranate 0.1671 5.984 Gooseberry 0.2935 3.407Grapes 0.1057 9.461 Feijoa 0.2074 4.822Figure 1 Graphic interpretation of Harrington’s individual desirability function given condition 1 d Cond and variable 1μD1 D2 D30.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20d1μ1I II III IVD1 D2 D30.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20d1μ1I II III IV354Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361point Di of Harrington’s individual desirability functiondetermined by Eq. (6) and corresponding to d1i; andd1i is the standard i-th critical (canonical) value d1 ofHarrington’s individual desirability function.The graphic interpretation of Harrington’s individualdesirability function corresponding to the conditionCondd1 is given in Fig. 1. For each value of d1i, wedetermined the corresponding values of μ1[Di ].As we can see, the 1μ range of definition includesfour domains separated by the critical values of μ1[Di ],where i = 1, 2, 3. By definition, domain IV includes those1 μvalues at which the fragmentation of the protopectincomplex makes no sense due to a low value of theindividual function of desirability.Domain III covers those 1μ values at which theindividual desirability function is large enoughfor protopectin fragmentation to make sense, butinsufficiently large to neglect non-uronide bioactivecomponents and isolate the products of fragmentation.In domains I and II, the individual desirabilityfunction is so large that the content of non-uronidebioactive components in plant tissue can be completelyignored.Based on the physical meaning of the boundaryconditions for 1μ, we established two individualboundary conditions that partially determined the nativepectin potential of plant tissue.Boundary condition I:– μ1 > μ1[D3 ] means the absence of the first individualpotential for protopectin fragmentation;– μ1 ≤ μ1[D3 ] means the presence of the first individualpotential for protopectin fragmentation.Boundary condition II:– μ1[D3 ] ≥ μ1 > μ1[D2 ] means the absence of the firstindividual potential for isolation of protopectinfragmentation products;– μ1 ≤ μ1[D2 ] means the presence of the first individualpotential for isolation of protopectin fragmentationproducts.Next, we determined the structure and propertiesof the second dimensionless individual criterion for theprotopectin fragmentation potential.The second independent variable was a complexoperator based on the mass fraction of protopectin in thetissue:2 100 2ωppϕ = = μ (26)where ϕ2 is the dimensionless operator of Harrington’sindividual desirability function; and μ2 is the seconddimensionless individual criterion for the protopectinfragmentation potential.Harrington’s individual desirability function wasexpressed as:( 20 21 2) ( 20 21 2)2d e e b b e e b b = − − + ⋅ϕ = − − + ⋅μ (27)Thus, the condition Condd2 that determined theindividual function was set as:221 2221 22: 0.35 ; : 0.65d 0.001 0.05Cond d dμ μ =  ⇔ ⇔  (28)Based on Condd2, we calculated the values of theconstant and the coefficient: 2b20 6.68 10= − ⋅ − andb21 = 18.179. The critical values of the μ2 criterion werecalculated as:( ) 20 2221ln ln[ ] iib dDbμ+ −  = − (29)where μ2[Di ] is the value of μ2 at the critical pointDi of Harrington’s individual desirability functioncalculated by Eq. (6) and corresponding to d2i; d2i is thestandard i-th critical (canonical) value d2 of Harrington’sindividual desirability function.Figure 2 Graphic interpretation of Harrington’s individual desirability function given condition Condd2 and variable 2 μD3 D2 D10.00.10.20.30.40.50.60.70.80.91.00.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20d2μ2IV III II ID1 D2 D30.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20d1μ1I II III IV×355Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361The graphic interpretation of Harrington’s individualdesirability function corresponding to the conditionCondd2 is presented in Fig. 2. For each value of 2i d , wecalculated the corresponding values of μ2[Di ].Just like with 1 μ , the μ2 range of definition includesfour domains separated by the critical values of μ2[Di ],where i = 1, 2, 3.By definition, domain IV covers those values of μ2at which the fragmentation of the protopectin complexmakes no sense. This led us to formulate the thirdindividual boundary condition:– μ2 < μ2[D3 ] means the absence of the secondindividual potential for protopectin fragmentation;– μ2 ≥ μ2[D3 ] means the presence of the secondindividual potential for protopectin fragmentation.We should note that fragmentation potentials I andII are categorical, i.e., if one of them is absent, the totalfragmentation potential is absent as well.Domains I, II, and III include such values of μ2 thatensure not only protopectin fragmentation, but alsothe isolation of fragmentation products. Based on thecanonical reference values of the individual desirabilityfunction, we formulated the fourth boundary condition:– μ2[D3 ] ≤ μ2 < μ2[D1] means the absence of thesecond individual potential for isolation of protopectinfragmentation products;– μ2 ≥ μ2[D1] means the presence of the secondindividual potential for isolation of protopectinfragmentation products.Similar to the first and the second fragmentationpotentials, the individual isolation potentials arecategorical.The third independent variable was a complexoperator based on the mass fraction of pectin substancesin the tissue:3 100 3ωpsϕ = = μ (30)where ϕ3 is the dimensionless operator of Harrington’sindividual desirability function; ωps is the totalamount of pectin substances, %; and μ3 is the thirddimensionless individual criterion for the protopectinfragmentation potential.In this case, the condition Condd3 that determined theindividual function was calculated as:331 3231 32: 0.40 ; : 0.65d 0.01 0.07d dCondμ μ =  ⇔ ⇔  (31)Based on expression (31), we calculated theconstant and the coefficient as 2b30 3.8367 10= − ⋅ − andb31 = 12.5788, respectively, and the critical boundaries ofμ3, as:( ) 30 3331ln ln[ ] iib dDbμ+ −  = − (32)where μ3[Di ] is the value of μ3 at the critical pointDi of Harrington’s individual desirability functioncalculated by (6) and corresponding to d3i; and d3i is thestandard i-th critical (canonical) value d3 of Harrington’sindividual desirability function.Figure 3 shows the graphic interpretation ofHarrington’s individual desirability function givenCondd3. For each value of d3i, we calculated thecorresponding values of μ3[Di ].Here, we can clearly see domain IV with no pectinpotential in the plant tissue.As a result, we formulated the fifth individualboundary 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 complexoperator based on the ratio of the mass fractions ofprotopectin and pectin substances in the tissue:4411pppsωϕω μ= =+ (33)where ϕ4 is the dimensionless operator of Harrington’sindividual desirability function; ωsp is the mass fractionof natively soluble pectin substances, %; and μ4 isthe third dimensionless individual criterion for theprotopectin fragmentation potential calculated as:4spppωμω= (34)Then, the conditionCondd4, which determined theindividual function, was calculated as:441 4241 42: 0.65 ; : 0.80d 2.50 1.25Cond d dμ μ =  ⇔ ⇔  (35)Based on expression (35), we calculated the constantand the coefficient (b40 = −0.3419, b41 = 4.1441).Based onCondd4, the critical boundaries of μ4 werecalculated as:( )41440 4[ ] 1ln ln iiD bb dμ = − −+ − (36)where μ4[Di ] is the value of μ4 at the critical pointDi of Harrington’s individual desirability functioncalculated by (6) and corresponding to d4i; and d4i is thestandard i-th critical (canonical) value d4 of Harrington’sindividual desirability function.Figure 4 shows the graphic interpretation ofHarrington’s individual desirability function givenCondd4, with d4i values corresponding to μ4[Di ] values.Based on the logical content of d4i and the numericalvalues of μ4[Di ], the range of definition can be dividedinto four domains that determine the fragmentationpotential of the protopectin complex and the isolationpotential of fragmentation products.According to Fig. 4, domain IV covers those valuesμ4 at which the mass fraction of water-soluble pectinexceeds that of the protopectin complex so muchthat there is practically no reason for its individualfragmentation. Thus, we determined the sixth boundarycondition as follows:– μ4 > μ4[D3 ] means the absence of the third individualpotential for protopectin fragmentation;– μ4 ≤ μ4[D3 ] means the presence of the thirdindividual potential for protopectin fragmentation.×356Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361Figure 3 Graphic interpretation of Harrington’s individual desirability function given condition Condd3 and variable μ3Figure 4 Graphic interpretation of Harrington’s individual desirability function given conditionCondd4 and variable 4 μ D1 D2 0.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 5 6 7 8 d1I II III D1 D2 D30.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15d4μ4I II III IVD1 D2 D30.00.10.20.30.40.50.60.70.80.91.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20d1μ1I II III IVD3 D2 D10.00.10.20.30.40.50.60.70.80.91.00.00 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 0.30d3μ3IV III II IFollowing the same pattern, we determined the seventhboundary condition (VII), namely:– μ4[D2 ] < μ4 ≤ μ4[D3 ] means the absence of thethird individual potential for isolation of protopectinfragmentation products;–μ4 ≤ μ4[D1] means the presence of the third individualpotential for isolation of protopectin fragmentationproducts.In addition, boundary conditions VI and VII arebased on:μ4 ≤ μ4[Di ] (37)where i = 3 for condition VI and, i = 2 for condition VII.However, μ4 can be expressed as:3 242sp ps pppp ppω ω ω μ μμω ω μ− −= = = (38)Then, given the presence of the third individualfragmentation potential:μ3 ≤ μ2 ⋅ (μ4[Di ]+1) (39)Thus, the third individual potentials of fragmentationand isolation are relative since they are involved inthe formation of respective total potentials indirectly,357Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361through expressions in which they act as one of thevariables.If we assume that there is a certain criterion spacewith coordinates μ1, μ2 and μ3, the pectin potentialof any plant material can be clearly determined asa geometrical location of the point M[μ1, μ2 , μ3 ]corresponding to the material under analysis.Based on the a priori assumption that1 ps i i 100λ ω ω=+Σ ≤ (40)we can establish the eighth boundary condition (VIII):the top boundary of the range of definition for allpossible values of M[μ1, μ2 , μ3 ] is determined by thefollowing basic proposition:μ3(top) = 1−μ1 ⋅μ2 (41)In addition, since a part cannot be larger than awhole, 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 forall possible values of M[μ1, μ2 , μ3 ] is determined by thesecond basic proposition:μ3(bot) = μ2 (44)The last formula is an expression of boundarycondition IX.RESULTS AND DISCUSSIONThus, according to boundary conditions VIII andIX, a set (A) of all points M[μ1, μ2 , μ3 ] can be defined asM[ 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 thepresence of pectin substances determines generalboundary conditions for defining a set of pointsM[μ1, μ2 , μ3 ] as the following hierarchy: “individualpectin potential → individual fragmentation potential ofthe protopectin complex → individual isolation potentialof protopectin fragmentation products”. Thus, the entireset of points M[μ1, μ2 , μ3 ] can be divided into foursubsets:– subset A1 characterized by the absence of a commonpectin potential in all the elements;– subset A2 where A2 ∩ A1 = ∅ and all the elements havea common pectin potential, but lack a common potentialfor protopectin fragmentation;– subset A3 where A3 ∩ A2 = ∅ and all the elementshave common pectin and protopectin fragmentationpotentials, but lack a common isolation potential forfragmentation products; and– subset A4 where A4 ∩ A3 = ∅ and all elementshave common pectin and protopectin fragmentationpotentials, as well as isolation potential forfragmentation products.By definition, the following is true for all the subsets:A1 ∩ A2 ∩ A3 ∩ A4 = ∅ (46)Based on the above, the existence of A1 correspondsto:μ2 ≤ μ3 < μ3[D3 ] (47)The area of definition for all A1 elements is partiallypresented in Fig. 6.The existence of subset A2 correspond2s to: 4 3 2 2 33 3 2 2 31 2 32 2 3 31 1 33 3 2 3 3( [ ] 1), [ ][ ], [ ]1, [ ][ [ ], [ ]D DD DDDD Dμ μ μ μμ μ μ μμ μ μμ μ μμ μμ μ ⋅ + ≥   < − ⋅ ≥ ≥ ≥  >  <2 4 3 2 2 31 1 33 3 2 2 31 2 32 2 3 31 1 33 3 2 3 3( [ ] 1), [ ][ ][ ], [ ]1, [ ][ ][ ], [ ]D DDD DDDD Dμ μ μ μμ μμ μ μμ μ μμ μ μμ μμ μ μ ⋅ + ≥  ≤  < − ⋅ ≥ ≥ ≥  >  <(48)Figure 7 shows a partial area of definition for all A2elements.The existence of A3 corresponds to: 1 2 4 21 2 1 1 3 2 32 4 3 2 2 3 1 3 1 1 22 3 2 2 ( [ ] 1)1 [ ] ( [ ] 1) [ ] [ ] [ [ ] [ DD D D D DD μ μ μμ μ μ μ μ μμ μ μ μ μ μ μμμ μ μ ≤ ⋅ +  − ⋅ ≤  ≥  > ≥  ⋅ + ≥ ≥ >  1 1 2  ≤ <2 4 21 2 1 1 3 2 2 132 4 3 2 2 3 1 3 1 1 222 3 2 2 1[ ]( [ ] 1)1 [ ] [ ]( [ ] 1) [ ] [ ] [ ][ ] [ ]DDD DD D D DD Dμ μμ μμ μ μ μ μ μμμ μ μ μ μ μ μμμ μ μ ≤ ⋅ +  − ⋅ ≤  ≥  > ≥  ⋅ + ≥ ≥ >   ≤ <(49)Figure 8 presents the area of definition for all A3elements.The existence of subset A4 corresponds to:1 2 1 1 23 22 4 2 2 2 11 [ ]( [ ] 1) [ ]DD Dμ μ μ μμ μμ μ μ μ − ⋅  ≤  > ≥   ⋅ +  ≥(50)The area of definition for all A4 elements is presentedin Fig. 9.Thus, the specific value M[μ1, μ2 , μ3 ] that shows itsbelonging to one of the subsets Ai (where i = 1, 2,3, 4) inthe zoned criterion space clearly determines the planttissue’s overall potential for protopectin fragmentation.Our approach to classifying plants as pectincontainingmaterials, which is based on a system ofcriteria and a zoned criterion space, has clear advantagesover existing methods due to its objectivity determinedby the boundary conditions.However, when analyzing this approach, we caneasily see that the μ j1 and μ j2 values correspondingto d j1 and d j2 in the conditions Condd j j=2,3,4 were seta priori, based on general assumptions regarding thedegree of acceptability of certain μ j values withinHarrington’s individual desirability functions inaccordance with the boundary (canonical) values of d.Yet, the conditions Condd j j=2,3,4 determine the coefficientsand constants, and, consequently, individual desirability358Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361functions, 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 generalizedalgorithm to determine the geometric location of planttissue in the zoned criterion space, or M[μ1, μ2 , μ3 ]belonging to one of the subsets (Fig. 10). We can usethis algorithm to assess any plant tissue’s potentialfor transformation of the protopectin complex, whichdetermines the influence of any technological impact onits pectin potential.The approach that we used to determine the criterionspace and boundary conditions for its zoning explicitlysuggests that this algorithm is universal for classifyingplant tissue or its derivatives as pectin-containingmaterials. Thus, the algorithm is applicable to any typeof plant material for which the μ1, μ2 and μ3 criteria canbe numerically expressed.CONCLUSIONTo sum up, our investigation showed the followingresults.1. We developed a system of criteria to assess thetransformation potential of the protopectin complex inplant tissue. This system is based on the geometricalFigure 6 Partial definition area for subset А1 Figure 5 Definition area of the criterion spaceFigure 7 Partial definition area for subset A2 Figure 8 Partial definition area for subset A3Figure 9 Partial definition area for subset A4359Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp. 348–361location of M[μ1, μ2 , μ3 ] – the point that correspondsto the material under analysis – in a zoned criterionspace with coordinates in the form of dimensionlessindividual criteria for protopectin fragmentationpotential.2. The dimensionless individual criteria forprotopectin fragmentation potential included the ratiobetween the mass fractions of biologically activecomponents and protopectin in plant tissue, the massfraction of the protopectin complex expressed inunit fractions, and the mass fraction of total pectinsubstances expressed in unit fractions.3. We established nine individual boundaryconditions, individual pectin potential, two individualfragmentation potentials, and three individual isolationpotentials for pectin substances, which altogetherdetermine a system of zoning the criterion space.4. The boundary conditions in the definition areafor a set of points M[μ1, μ2 , μ3 ] had the followinghierarchy: individual pectin potential → individualStart𝜔𝑠𝑝; 𝜔𝑝𝑝; 􀷍 𝜔𝑖𝜆𝑖=1𝜇1; 𝜇2; 𝜇3; 𝜇4𝜇1[𝐷𝑖]|𝑖=2,3; 𝜇2[𝐷𝑖]|𝑖=1,3; 𝜇3[𝐷𝑖]|𝑖=3; 𝜇4[𝐷𝑖]|𝑖=2,3𝜇3 < 𝜇3[𝐷3]𝜇1 > 𝜇1[𝐷3]𝜇2 < 𝜇2[𝐷3]𝜇3 ≥ 𝜇2 ∙ (𝜇4[𝐷3] + 1)𝜇2 < 𝜇2[𝐷1]𝜇1 > 𝜇1[𝐷2]𝜇3 ≥ 𝜇2 ∙ (𝜇4[𝐷2] + 1)Finishprotopectin complex’fragmentation withproduct isolationwithout protopectincomplex’ fragmentationprotopectin complex’fragmentation withoutproduct isolationpecticpotential is absentyes nono nono yesyesnoyes noyes noyesnoА1А2А3 А4Figure 10 Algorithm for plant tissue classification according to protopectin fragmentation potential based on the geometriclocation in the zoned criterion space360Kondratenko V.V. et al. Foods and Raw Materials, 2020, vol. 8, no. 2, pp.348–361fragmentation potential of the protopectin complex→ individual isolation potential of protopectinfragmentation products.5. We developed an algorithm to classify planttissues according to protopectin fragmentation potentialbased on the geometric location in the zoned criterionspace.CONTRIBUTIONAll the authors were equally involved in writing themanuscript and are equally responsible for plagiarism.CONFLICT OF INTERESTThe authors state that there is no conflictof interest.