Statistical Modeling of Chlorinated Chemical Compounds Bioactivity

VLADIMIR MUKHOMOROV

Physical Department

State Polytechnic University

St. Petersburg

RUSSIA

Abstract: - A general regression equation based on physical concepts of the behavior of a polar molecule in a

condensed medium is derived. The regression equation makes it possible, from a unified standpoint, to

statistically significantly explain the toxicity of both chlorine-substituted benzenes and saturated and

unsaturated chlorinated hydrocarbons. Statistically significant explanatory molecular features that determine

the bioactivity of drugs have been identified.

Key-Words: - Regression, significance, information, quality criteria, collinearity, toxicity, intermolecular,

electronic, pseudopotential, chlorinated chemical compounds

Received: May 26, 2021. Revised: April 15, 2022. Accepted: May 12, 2022. Published: June 6, 2022.

1 Introduction

Much attention has recently been paid to finding

various quantitative relationships that link variations

in the molecular structure of chemical compounds to

their biological activity. For these purposes, either

abstract statistical models are used (leaving the

mechanism of biological activity undisclosed) or

assumed physico-chemical ideas about the possible

behaviour of chemical compounds in the biosystem.

This article analyzes the toxic effects of

chlorobenzene derivatives, as well as saturated and

unsaturated chlorine-containing compounds. Using

the methods of the theory of intermolecular

interactions, the corresponding regression

relationships will be derived here for the purposes of

statistical analysis of the relationship between the

structure of a molecule and its biological activity.

2 Problem Formulation

In accordance with modern concepts, the biological

activity of chemical compounds is determined by

their physicochemical properties at the macroscopic

level (solubility, distribution, permeability), as well

as at the microscopic level (electronic characteristics

of molecules). In this regard, it can be assumed that

the biological effect is equally determined by two

circumstances: the transport of the molecule to the

site of action and the physicochemical interaction of

the molecule with the receptor. Attempts to obtain

appropriate regression equations that take into

account different factors have been repeatedly

discussed in the literature [1,2]. The authors of the

papers [3,4] point out the difficulties in the

physicochemical interpretation of the observations

used in these studies.

3 Problem Solution

3.1 Chlorinated benzene derivatives

The Hansch model [5,6] has been the most widely

used in recent years. This model relates the

bioactivity of chemical compounds to their

lipophilic characteristics. In many practical cases,

this model has proved useful. Therefore, let us

check whether the bioactivity (average lethal doses

of benzene chlorine derivatives for white rats upon

oral administration [7]) is really related to the

partition coefficient P of the substance in the

octanol–water system. We use the well-known

Hansch equation

A = B0 + B1lgP + B2(lgP)2, (1)

here A = 1000/LD50 is bioactivity, B0, B1, and B2 are

some unknown parameters that are defined by

minimizing the squared deviation of function values

(1) from known experimental values. Toxicity and

lgP values for a number of substituted benzenes are

given in Table 1.

Using equation (1), the coefficient of determination

R2 = 0.237 was determined. This coefficient

characterizes the magnitude of the statistical

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

115

Volume 21, 2022

relationship between activity A and distribution

parameters for immiscible solvents. That is, the

“explaining” ability of the model is only 23.7%. The

statistical significance of the multiple correlation

coefficient can be tested by using the following

inequality [8]: t = |R|(N – m – 1)0.5/(1 – R2)0.5 = 1.67

< t0.05cr(f = 9) = 2.26; where m is the number of

explanatory variables, N is the sample size.

Consequently, it must be recognised that there is no

reliable relationship between regression (1) and the

observed toxicity at the 95% confidence level.

Comparison of the multiple correlation coefficient R

= 0.49 with the critical value R0.05cr(N – m – 1 = 9;

m = 2) = 0.697 [9] also indicates that the correlation

coefficient R is insignificant at the significance level

α = 0.05. Moreover, as the analysis showed, the use

of various modifications of equation (1) (see, for

example, [10]), including the use of the Hammett

constants ϭ, also does not allow one to establish a

relationship between changes in the structure of the

molecule and the variation of the biological

response. Apparently, the relationship between the

molecular structure and the bioactivity of a chemical

compound must be sought based on other properties

of this series of compounds. In molecular

pharmacology, it is known that the biological action

of a chemical compound depends on its ability to

accumulate in certain areas of the body through

interaction with sensitive local biostructures of the

body.

Analysis of the relationship between various toxicity

indicators of chemical compounds and their

physicochemical properties showed the following

[11]. The greatest number of correlations is found

with the properties of low-molecular weight

compounds, which are determined at the electronic

level and are related to the energy of intermolecular

interaction of molecules [12,13].The presence of

long-range components in the energy of the

intermolecular interaction must lead to a

concentration gradient of chemical compounds. This

contributes to the emergence of a diffusion flow of

low molecular weight chemical compounds directed

towards the active center.

The process of binding exogenous molecules in the

body can approximately determine the additive

components of pairwise intermolecular interactions.

Difficulties in consistently taking into account the

contributions to the intermolecular interaction are

due to the variety of types of pair interactions of

molecules, as well as the possibility of correctly

taking into account the influence of the condensed

phase on these interactions. Intermolecular

interactions are usually divided into two groups

according to their radius of action and relative

strength: specific and non-specific (universal). The

first group includes anisotropic pairwise quasi-

chemical bonds (donor-acceptor complexes,

hydrogen bonds) that arise when the electron shells

of interacting molecules overlap markedly.

Nonspecific interactions include various

electrostatic interactions, as well as short-range

dispersion forces. These interactions are determined

not only by the individual properties of individual

chemical compounds, but also by the properties of

the biosubstrate, that is, the condensed phase in

which exogenous chemical compounds are

distributed. The proposed mathematical model

should highlight the additive components of the

interaction, that is, orientation interactions,

polarization and short-range contributions (on a

molecular scale). These contributions are related to

the dipole moment, electronic polarizability,

ionization potential, and also to the position of one-

electron energies MO on the energy scale of an

isolated molecule. This approach makes it possible

to establish the existence of possible causal

relationships between molecular features and

bioresponse.

The analyzed series of chlorobenzene derivatives

(Table 1) is interesting from the point of view that in

this case it is possible to move away from the

problems associated with the conformational

transitions of molecules. In addition, these

molecules have similar sizes and are not expected to

be involved in metabolic transformations.

It is known [16,17] that chlorinated compounds

have good acceptance properties. This allows them

to participate in the formation of donor-acceptor

molecular complexes due to electron transfer. The

change in total energy (∆E) when a bond is formed

between atom s of the donor molecule and atom t of

the acceptor molecule can be written as follows

ΔE = - qsqt/(κsRst) +

2ΣmΣn(CsmCtnΔβst)2/(εm – εn). (2)

Here the summation occurs over the m occupied

molecular orbitals (MO) of the donor and over the n

vacant MO of the acceptor; εm and εn are the

energies of single-particle MO of the donor and

acceptor, respectively; ∆βst is the change in the

resonance integral of the interacting atoms s and t at

the distance Rst between the atoms; Csm and Ctn are

the expansion coefficients of MO in atomic orbitals;

κs is the static permittivity of the condensed

medium.

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

116

Volume 21, 2022

Table 1

Molecular parameters of chlorobenzene derivatives and their mean lethal doses (LD50) for white rats

after oral administration of drugs.

№

Chemical compounds

lgP

εnb0, eV

I1*), eV

μ1, D, [14]

α1**),1024cm3

Z,

arb.

units

H, bits

Aexp, 1000/LD50

[14]

Amod, (15)

1

Chlorobenzene

2.84

-0.971

9.15

1.69

13.2

3.00

1.33

0.303

0.248

2

p-Dichlorobenzene

3.39

-1.441

9.11

0

15.0

3.50

1.46

0.398

0.476

3

o-Dichlorobenzene

3.39

-1.356

9.23

2.51

15.9

3.50

1.46

0.468

0.788

4

1,2,4,5-Tetrachlorobenzene

4.89

-2.097

9.31

0

19.7

4.50

1.46

0.667

1.036

5

2,4,6-Trichlorophenol

3.06

-1.751

9.02

1.62

19.0

4.00

1.78

1.299

0.900

6

1,2,4-Trichlorobenzene

4.13

-1.764

9.26

1.25

18.3

4.00

1.50

1.323

0.847

7

3,4-Dichloraniline

2.69

-1.292

8.24

4.16

17.8

3.43

1.73

1.429

1.405

8

p-Nitrochlorobenzene

2.39

-3.112

10.21

2.52

16.5

3.71

1.99

1.802

2.291

9

m-Nitrochlorobenzene

2.46

-3.077

10.00

3.38

17.9

3.71

1.99

2.326

2.572

10

o-Nitrochlorobenzene

2.53

-2.987

9.95

4.25

16.7

3.71

1.99

2.949

2.900

11

2,4-Dinitrochlorobenzene

2.45

-3.657

10.86

3.29

19.1

4.25

2.11

3.571

3.030

12

2,3,5,6-Tetrachloronitrobenzene

4.55

-3.576

9.86

5.34

24.1

5.00

1.99

4.000

4.041

*) Dipole moments of molecules and their ionization potentials were calculated using the MINDO/3 quantum mechanical

method. **) The polarizabilities of the molecules were determined using the Lefevre additive scheme [15].

The surrounding dielectric medium is considered as

structureless and continuous, which is characterized

by the dielectric constant κs. The first term in

equation (2) determines the electrostatic interaction

between atoms, which have electronic charges qs

and qt. Electrostatic forces favor the interaction of

donor and acceptor atoms, but usually are not

decisive for the stabilization of the complex. In

highly polar solvents, electrostatic interactions are

significantly weakened. The second term defines

quasi-chemical binding, i.e. it characterizes the

partial electron transfer from the donor MO to the

acceptor, thereby stabilizing the molecular complex.

The donor-acceptor mechanism arises when the free

orbital of the acceptor overlaps with the filled

orbital of the donor or donor group of atoms. The

donor is assumed to have a lone pair of electrons.

For example, a nitrogen atom has a lone pair of

electrons in the state 2s2. When two molecules

approach each other, the lone pair of electrons is

shared between the two molecules.

This generalization of electrons is accompanied by

the formation of bonds between molecules. The

interaction between the donor and acceptor leads to

a decrease in the energy of the ground state of the

entire system below the initial levels of the donor

and acceptor. The measure of the acceptor activity

of a chemical compound having a closed electron

shell is the position of the lowest free MO (εnb0).

Moreover, the acceptor properties are stronger, the

lower the level εnb0. Index zero indicates that one-

electron energy corresponds to an isolated molecule

in vacuum.

The quantum-chemical method was used to

determine the numerical values of one-electron MO

energies CNDO/S’ [18]. The experimental values of

the lengths of interatomic bonds, bond angles of

molecules, for all chemical compounds analyzed

here, were taken from the reference book [19].

In a condensed polar medium, under the influence

of the electrostatic field of the dipole molecules of

the environment, the electronic levels are shifted

relative to their position in an isolated molecule.

The macroscopic electric field Eeff acting on a

molecule in a condensed medium differs from the

average macroscopic field. This is due to the effect

of polarization of the dielectric in an external field,

as well as due to the action of the reactive field of

the polar molecules of the dielectric medium. The

induced (reactive) electric field acts on the field

source, changing the electronic distribution of the

molecule (i.e., self-action of the polar molecule

occurs). The presence of a polarizable medium

between interacting molecules can significantly

change their total potential energy.

As is known [20], the reactive field of L.Onsager

[21] acting on a molecule is proportional to its

dipole moment. To determine the effect of a reactive

field on the electronic states in a molecule, we use

the concepts developed in molecular spectroscopy

of the condensed state [22]. Intermolecular

interactions can significantly affect the optical

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

117

Volume 21, 2022

spectra of molecules, shifting the maxima of the

absorption and emission bands, changing the

intensity of the bands, and new spectral frequencies

may appear.

Within the framework of the continuum theory of

the reaction field, the one-electron energy εnb, under

the condition of thermodynamic equilibrium of the

molecule with the environment, will be determined

as follows:

εnb = εnb0 – fRμ12/(2a3) + α1fR2μ12/(2a6) –

3I1I3α1α3(n32–1)/[2(I1 + I3)a6(n32+2)]. (3)

Here, the reaction field and dispersion interactions

are taken into account. fR = 2(κs–1)/(2κs + 1) is the

electric field factor of the reaction of a point dipole

for a polar medium with a static permittivity κs; a,

μ1, α1 and I1 are the effective size of a molecule of a

low molecular weight exogenous chemical

compound commensurate with the average radius of

the molecule, its dipole moment, as well as the

average static electronic polarizability and the first

ionization potential of the molecule, respectively.

In a condensed medium, each molecule is under the

influence of a combination of surrounding

molecules. This is partly taken into account by the

factor fR, which depends on the macroscopic

properties of the condensed medium. Let us agree

that index 1 refers to the molecule of an exogenous

chemical compound, index 2 refers to the

biosubstrate molecule with which the molecule of

the exogenous substance interacts. Index 3 refers to

a polar dielectric medium, the optical refractive

index of which is equal to n3. The second and third

terms in equation (3) describe the interaction of the

dipole moment of the molecule with the field of the

electrostatic reaction of the polar dielectric medium

[22] and, thus, the effect of polarization of the

molecule by this field is taken into account. The

third term in (3) characterizes the effect of short-

range dispersion interactions (in the London

approximation) on the MO levels of an exogenous

chemical compound in a condensed medium.

On a macroscopic scale, the role of the reaction field

manifests itself in the fact that the liquid is

compressed in the region of the surrounding

molecule, thereby increasing the potential energy of

the polar liquid medium. This is one of the reasons

for the increase in the boiling point (Tb) and

decrease in the melting point (Tm) of polar liquids.

Therefore, it is natural that the correlation

coefficients between the bioresponse and each of the

parameters Tb, Tm and μ1 are close in magnitude to

each other [11]. However, in this case, the

regression equations are not informative enough,

since it is not clear which approximately additive

contributions of intermolecular interactions should

be taken into account in each particular case.

One can make some assumptions about the

molecular properties of the local region with which

the exogenous molecule is associated, if the main

contributions are known. For example, if dispersion

forces make a dominant contribution, then a local,

biological object must have high polarization

properties.

When a low molecular weight chemical compound

approaches a receptor, the molecule enters into a

pair interaction with it. In the dipole approximation,

the efficiency of this interaction is determined by

the following additive physical components:

1) dipole - dipole interaction, which after averaging

over all orientations of molecules at temperature T

has the following form

Edip = - 2μ12μ22/(3κsR06kBT), (4)

that is, the attraction between the dipoles depends

on the temperature;

2) inductive interaction

Eind = - (α2μ12 + α1μ22)/(κs2R06); (5)

3) dispersion interaction, which, in the London

approximation, can be written in the following form:

Edisp = - 3α1α2I1I2/[2R06(I1 + I2)]. (6)

Approximate formulas (3) - (6) make it possible to

write the pair interaction of molecules in terms of

the properties of individual molecules. Here R0 is

the effective distance between the interacting

molecules; αi is the isotropic electronic

polarizability of the i-th molecule; μi is the dipole

moment of the i-th molecule; kB is the Boltzmann

constant; Ii is the first ionization potential of the i-th

molecule. It should be borne in mind that the

assumption of the additivity of intermolecular

interactions is not sufficiently rigorous [23].

However, for solving practical problems, the

approximation of interaction additivity is generally

accepted. The possibility of using simplified

formulas makes it possible to significantly expand

the scope of the theory of intermolecular

interactions.

Studies have shown [20,24] that formulas (4) - (6)

are applicable to real systems. These

approximations make it possible to correctly

indicate the changes in the potential energy of the

interaction of molecules as a function of the distance

between them and the individual properties of the

molecules. Before writing the final equation that

determines the change in the energy of the system,

taking into account specific and nonspecific

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

118

Volume 21, 2022

interactions, let's simplify equation (2). For

homologous series of chemical compounds or

related compounds, which interact with the same

donor, the electron-accepting properties of the

molecule mainly depend on the position on the

energy scale of the lower free molecular orbital εnb

of the acceptor: ∆Ed-a = f(εnb). For the homological

series of chemical compounds, the change in

interaction energy is approximately proportional to

the energy εnb: ∆Ed-a ≈ εnb. In general, the equation

that determines the value of the stabilization energy

of the complex of a low-molecular chemical

compound plus a receptor will have the following

form:

ΔEd-a = k0 + kεnb0 + μ12{-kfR/(2a3) + kα1fR2/(2a6) –

2μ22/(3κsR06kBT) – α2/(κs2R06)}+ α1{–μ22/(κs2R06) –

3α2I1I2/(2R06(I1 + I2))} –

[3α1I1I3/(2a3(I1 + I2))]∙[(n32 –1)/(n32 + 2)]. (7)

Here k and k0 are some numerical coefficients. For

the purposes of regression analysis, this equation

can be simplified. It is known that the inductive

interaction Eind << Edisp and therefore in the fourth

term of the series (7) the first term in the curly

bracket can be ignored in comparison with the

second term. In what follows, we will assume that

the molecular parameters that relate to the

biosubstrate and the polar medium are constant for

the entire range of chemical compounds.

In this study, the active center of the biophase with

which the exogenous molecules interacts is not

specified. Therefore, for certainty we will assume I2

≈ I3 ≈ 10 eV. This value of the ionization potential

corresponds to most organic molecules [25,26].

Since the main goal is to obtain a regression

equation, this approximation is quite satisfactory.

Thus, equation (7) can be reduced to the following

multifactorial regression equation with three

explanatory variables that enter the equation

additively and have a joint simultaneous effect on

the resulting trait:

A ≡ 1000/LD50 = B0 + B1εnb0 + B2μ12 +

B3α1I1/(I1 +10). (8)

For the convenience of presenting the statistical

material, in what follows we introduce the following

notation: x1 = εnb0, x2 = μ12 и x3 = α1I1/(α1 + 10).

Next, the problem is reduced to estimating the

multiple regression coefficients Bi from the known

results of sample observations.

The statistics of the populations A, x1, x2 and x3 will

be as follows:

A = 1000/LD50: N = 12, Aav = 1.71 ± 0.36; 95%

confidence interval: 0.91 - 2.51; Amin = 0.303, Amax

= 4.00, SA = 1.266, τmin = 1.12 < τmax = 1.82 <

τ0.05cr,2(N) = 2.387 < τ0.05cr,1(N) = 2.523; Wilk-

Shapiro normality test: W = 0.910 > W0.05cr(N) =

0.859, David-Hartley-Pearson normality test:

U10.05cr(N) = 2.800 < U = [(Amax – Amin)/SA] =

2.92 < U20.05cr(N) = 3.910, representativeness of the

sample size: Nrepr = 10;

x1: N = 12, x1av = - 2.26 ± 0.28; 95% confidence

interval: (-2.87, -1.64); εnb0,min = -3.67, εnb0,max = -

0.97, Sx1 = 0.964, τmax = 1.34 < τmin = 1.46 <

τ0.05cr,2(N) = 2.387 < τ0.05cr,1(N) = 2.523; Wilk-

Shapiro normality test: W = 0.895 > W0.05cr(N) =

0.859, David-Hartley-Pearson normality test:

U10.05cr(N) = 2.800 ≈ U= [(εnb0,max – εnb0,min)/Sх1] =

2.791 < U20.05cr(N) = 3.910; Nrepr = 10;

x2: N = 12, x2av = 8.82 ± 2.54; 95% confidence

interval: 3.23 - 14.41; μ12,min = 0, μ12,max = 28.52, Sx2

= 8.80, τmin = 1.00 < τmax = 2.24 < τ0.05cr,2(N) =

2.387 < τ0.05cr,1(N) = 2.523; Wilk-Shapiro normality

test: W = 0.885 > W0.05cr(N) = 0.859, David-Hartley-

Pearson normality test: U10.05cr(N) = 2.800 < U =

[(μ12,max – μ12,min)/Sх2] = 3.25 < U20.05cr(N) = 3.910;

Nrepr = 10;

x3: N = 12, x3av = 8.43 ± 0.42; 95% confidence

interval: 7.75 - 9.58; x3min = 5.276, x3max = 11.965,

Sx3 = 1.744, τmin = 1.81 < τmax = 2.03 < τ0.05cr,2(N)

= 2.387 < τ0.05cr,1(N) = 2.523; Wilk-Shapiro

normality test: W = 0.975 > W0.05cr(N) = 0.859,

David-Hartley-Pearson normality test: U10.05cr(N) =

2.800 < U = [(x3max – x3min)/Sх3] = 3.83 = U20.05cr(N)

= 3.910; Nrepr = 10. (9)

Since the sample size is limited, before analyzing

the regression (8), we perform the following

procedure. Let's analyze a regression that uses only

one explanatory variable, such as the variable x2:

A1(x2) = a0 + a1 x2.

For this regression, we get the following statistics:

N = 12, m1 = 1; R = 0.80 ± 0.11, |R*| = 0.82 >

R0.05cr(N – 2) = 0.576; correlation coefficient

significance test based on the Fisher normalizing z-

transform (with Hotelling corrections taken into

account): uH = 1.07 > u0.05(N) = z0.975∙(N – 1)-0.5 =

0.591; RMSE = 0.794; the minimum sample size

sufficient for the reliability of the correlation

coefficient: N0.05min = 6; a0 = 0.17 ± 0.09, a1 = -1.01

± 0.26, |t(a1)| = 3.8 > t0.05cr(N – 2) = 2.228 > t(a0) =

1.93; unexplained regression residuals (perturbing

variable) are normally distributed: Wilk-Shapiro

test: W = 0.938 > W0.05cr(N) = 0.859; F = 17.57 >

F0.05cr(f1 = 1;f2 = 10) = 4.96.

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

119

Volume 21, 2022

Here R* = R∙[1 + 0.5∙(1 – R2)/(N – 3)] is the adjusted

correlation coefficient. Thus, there is a significant

relationship between the explanatory variable x2 and

bioresponse. Checking the relationship of regression

residuals δA1 with the explanatory variable x1 also

indicates the presence of a significant correlation

between them:

δA1(x1) = a01 + a11 x1,

N = 12, m1 = 1; R = - 0.74 ± 0.14, |R*| = 0.76 >

R0.05cr(N – 2) = 0.576; correlation coefficient

significance test based on the Fisher normalizing z-

transform (with Hotelling corrections taken into

account): uH = 0.92 > u0.05(N) = z0.975∙(N – 1)-0.5 =

0.591; RMSE = 0.533; the minimum sample size

sufficient for the reliability of the correlation

coefficient: N0.05min = 7; a01 = -1.31 ± 0.41, a11 = -

0.58 ± 0.17, |t(a11)| = 3.48 > t0.05cr(N – 2) = 2.228;

unexplained regression residuals (perturbing

variable) are normally distributed: Wilk-Shapiro

test: W = 0.944 > W0.05cr(N) = 0.859; F = 12.13 >

F0.05cr(f1 = 1;f2 = 10) = 4.96.

It follows that the regression δA1(x1) is also

statistically significant. That is, the unexplained

residuals of regression A1(x2) correlate with another

explanatory variable, namely variable x1. Further

verification of the relationship between the

regression residuals δA1(x1) and the explanatory

variable x3 showed that the residuals are not

associated with the molecular index characterizing

the dispersion interaction: R = 0.31 < R0.05cr(N – 2) =

0.576, F = 1.07 < F0.05cr(f1 = 1;f2 = 10) = 4.96.

Obviously, this contribution to the interaction of

molecules is insignificant for the analyzed sample.

It follows from inequalities (9) that at a significance

level of 5%, the populations A, x1, x2, and x3 are

homogeneous and normally distributed. The

homogeneity of the analyzed data was checked

using the τ-criterion [27,28]. The following statistics

were obtained for regression (8):

N = 12, m1 = 3 – number of explanatory

variables; multiple correlation coefficient: R1 =

0.966 > R0.05cr(m1; N – m1 – 1) = 0.777, multiple

determination coefficient: R12 = 0.933, R1*2 = 0.91;

standard error of the regression estimate: SA = 0.381;

B0 = - 1.29 ± 0.82, B1 = -0.78 ± 0.18, B2 = 0.06 ±

0.02, B3 = 0.09 ± 0.12; |t(B1)| = 4.27 > t(B2) =

3.51 > t0.05cr(f = N – m – 1) = 2.306 > t(B3) = 0.70;

F = 37.29 > F0.05cr(f1 = 3; f2 = 8) = 4.07; Σ1 =

1.1586; AIC1 = -1.837, SC1 = -1.5093, SS1 =

0.1196. (10)

Here Σ1 is the sum of squared residuals; R0.05cr(m1; N

– m1 – 1) is the critical value of the multiple

correlation coefficient [9], which determines the

lower acceptable limit of the degree of association

between the variations of the resulting attribute and

all explanatory variables.

Statistics (10) contains information quality criteria

for the linear regression equations of Akaike [29]

and Schwartz [30], as well as the alternative ratio SS

= Σ0.5/(N – m). For regression residuals, the Wilk-

Shapiro test would be as follows: W = 0.975 >

W0.05cr(N) = 0.859. The information quality criteria

for the regression equation are defined as follows:

AIC = (2m/N) + ln(Σ/N),

SC = [(m + 1)lnN]/N + ln(Σ/N) . (11)

The assessment of the significance of the multiple

determination coefficient (10) is carried out using

the F-statistics: F = R2∙(N – m – 1)/m/(1 – R2). Since

F > F0.05cr (see Eq.(10)), it can be assumed that the

multiple coefficient of determination is reliably

different from zero with probability 1 – α = 0.95,

and the explanatory variables reliably explain

variations in bioactivity. The Bi regression

coefficients are significantly greater than zero if

t(Bi) > tcr at significance level α and number of

degrees of freedom f = N – m1 – 1 at the two-sided

critical region. Therefore, the coefficient B3 in

Eq.(10) is not statistically reliable. Regression (8)

explains 93.3% of the variability in bioactivity.

Only 6.7% of unexplained variations can be

attributed to unaccounted for factors or random

variations in the original data.

The importance of the participation of each of the

independent explanatory variables in assessing the

variability of the resulting sign is characterized by

standardized regression coefficients. The

standardized regression coefficients Bi* are related

to the normal regression coefficients (8) by the

following relationships:

B1* = B1∙Sx1/SA = - 0.596, B2* = B2∙Sx2/SA = 0.405,

B3* = B3∙Sx3/SA = 0.098. (12)

On a natural scale, the regression coefficients are

dimensional quantities. However, the standardized

coefficients are dimensionless and this makes it

possible to perform their quantitative comparison.

Knowledge of standardized coefficients makes it

possible to determine the proportion of explanatory

variables involved in explaining the variability of

the resulting sign. An approximate ratio for the

multiple coefficient of determination can be used to

obtain information about the comparative influence

of individual variables [31]:

Rappr 2 = B1*∙rx1,A + B2*∙rx2,A + B1*∙rx3,A =

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

120

Volume 21, 2022

0.535 + 0.321 + 0.075 = 0.931. (13)

Here rx1,A = -0.897, rx2,A = 0.798 and rx3,A = 0.770

are the pairwise correlation coefficients between

each explanatory variable and the observed

bioactivity. From relation (13) it follows that the

greatest contribution to the explanation of the

variability in the toxicity of chemical compounds

comes from the variables x1 (53.5%) and x2 (32.1%).

The equity participation of variable x3 is very

insignificant and amounts to only 7.5%. The

approximate value of the coefficient of

determination (13) is very close to the value of R12 =

0.933 (see Eq.(10)).

The adjusted (unbiased) coefficient of determination

is determined from the following:

R*2 = 1 – (1 – R2)·(N – 1)/(N – m – 1). (14)

An adjusted coefficient of determination, R*2 is

applied so that models with different numbers of

explanatory factors can be compared.

Thus, at the 95% confidence level, a very strong

relationship can be assumed between toxicity and

the explanatory variables x1 and x2. The coefficient

of determination R12 = 0.933 (10) indicates what

portion (in this case, 93.3%) of the total variance of

the bioresponse function is explained by the factors

x1, x2, and x3. Only 6.7% of the total variance cannot

be explained by the model (the uncertainty factor is

0.067) and appears to be due to unaccounted for

variables or random deviations in the original

sample. The values of coefficients B1* and B2*

reflect the significant dependence, at the 95%

confidence level, of the resultant variable on the

explanatory factors x1 and x2. At the same time, the

intermolecular dispersion interaction (~ x3) is of

minor importance (not significant at the chosen

significance level α), since t(B3) < tcr (two-sided

hypothesis evaluation). The difference of the

coefficient B3 from zero can be attributed to random

fluctuations in the original sample. Therefore,

nothing definite can be said about the influence of

the dispersion interaction on the resultant attribute.

Features that have low information content (i.e.,

"weight") can be excluded from further analysis.

The choice of independent explanatory variables is a

process of successive refinement of the initial

hypothesis. The following steps can be

distinguished in this process: formation of a primary

hypothesis (Eq. (8)) about the set of independent

variables; analysis of structural relationships;

narrowing of features and selection of significant

variables for modeling.

Since the influence of dispersion interactions is

insignificant, equation (8) can be replaced by the

following reduced two-factor regression equation:

A ≡ 1000/LD50 = B0 + B1x1 + B2x2, (15)

N = 12, m2 = 2, B0 = -0.76 ± 0.29, B1 = -0.86 ± 0.14,

B2 = 0.06 ± 0.02; |t(B1)| = 6.10 > t(B2) = 3.98 >

|t(B0)| = 2.65 > t0.05cr (f = N – m2 – 1) = 2.26; R2 =

0.968 > R0.05cr (2;9) = 0.697, R22 = 0.937, R2*2 =

0.933; standard error of the regression estimate: SA

= 0.370; F = 59.04 > F0.05cr(f1 = m2; f2 = N – m2 – 1)

= 4.26; Σ2 = 1.2294; AIC2 = -1.9450, SC2= -1.6572,

SS2 = 0.1109. (16)

Regression residuals are normally distributed. Wilk-

Shapiro normality test: W = 0.940 > W0.05cr(N) =

0.859.

If the ratio F many times (for example, not less than

four times) exceeds the tabular value, then such a

regression, according to [32], has predictive

properties. Additional information about the

significance of variable x3 can be obtained by

analyzing the relationship between the regression

residuals (15) and variable x3. As the analysis

showed, the correlation coefficient is insignificant:

R = 0.16 < R0.05cr(N – 2) = 0.576, that is, the

explanatory variable x3 is not related in any way to

the unexplained regression residuals (15).

Therefore, this variable does not really need to be

included in the regression equation. A comparison

of the information tests AIC1 and AIC2 for

regressions (8) and (15) shows that the quality of the

reduced regression (15) is higher than for regression

(8), although the number of explanatory variables

has decreased. Similar relations hold for the tests

SC1 = -1.5093 and SC2 = -1.6572. The values of the

Akaike and Schwarz tests are associated with the

ratios SS: SS1 = Σ10.5/(N – m1) = 0.1196 and SS2 =

Σ20.5/(N – m2) = 0.1109. The use of AIC, SC and SS

tests is justified because the R2 criterion may not be

informative. The fact is that for a model with a large

number of explanatory variables, the criterion R2

will always be no less than for a model with a

smaller number of explanatory variables. The

presence of collinearity between explanatory

variables can significantly distort the relationship

between the information tests for full regression and

reduced regression.

If we analyse the residuals δA of a simple linear

regression that depends on only one explanatory

variable x1: A ≡ 1000/LD50 = B0 + B1x1 (|R| = 0.90 >

R0.05cr(N – 2) = 0.576; F = 41.2 > F0.05cr(f1 = 1;f2 =

10) = 4.96), then appears that the residuals δA of the

regression are significantly correlated with the set of

variables x2: r = 0.66 > R0.05cr(N – 2) = 0.576, F =

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

121

Volume 21, 2022

7.76 > F0.05cr(f1 = 1; f2 = 10) = 4.96. Thus, for the

regression equation (15), both explanatory variables

x1 and x2 are significant. However, it is necessary to

check whether these variables are not significantly

correlated. Collinearity between explanatory

variables can lead to misjudgment of the impact of

variables on the outcome variable because the

explanatory variables are related. The presence of

collinearity between explanatory variables can

significantly distort the test relationships for full

regression and reduced regression. Checking the

interconnectedness of the explanatory variables x1

and x2 is carried out as follows:

N = 12; x1(x2) = b0 + b1x2, r1,2 = -0.564 ± 0.216,

|r1,2| = 0.564 < R0.05cr(N – 2) = 0.576; |t(b1)| = 2.16 <

t0.05cr(N – 2) = 2.228; b0 = -2.80 ± 0.81, b1 = -6.15 ±

2.38; the minimum sample size sufficient for the

reliability of the correlation coefficient: N0.05min = 11;

F = 4.67 < F0.05cr(f1 = 1; f2 = 10) = 4.96. (17)

Since F < Fcr, the relationship between x1 and x2 is

not significant at the 95% confidence level. For

small sample sizes (N ≤ 15), the best estimate of the

correlation coefficient is the adjusted correlation

coefficient [33]:

r* = r∙[1 + 0.5∙(1 – r2)/(N – 3)]. (18)

For regression (15), the residuals are normally

distributed (16). In this case, a more accurate

quantification of collinearity can be used, as

suggested by Farrar and Glauber [34]. Farrar-

Glauber test, has a chi-square distribution with f =

m(m – 1)/2 degrees of freedom:

χ2 = - (N – 1 – (2m2 + 5)/6)∙ln(r1,1∙r2,2 – r2,1∙r1,2)

= 3.64 < χ0.052,cr(f = 1) = 3.841. (19)

Here m2 = 2 is the number of explanatory variables.

Since χ2 < χ2,cr, then the hypothesis of the

independence of the explanatory variables does not

contradict the original data. The absence of a

significant relationship between the features εnb0 and

μ12 is also indicated by the inequality:

t = |r1,2|∙(N – 2)0.5/(1 – r1,22)0.5 =

2.16 < t0.052,cr(N – 2) = 2.23. (20)

That is, the correlation coefficient |r1,2| statistically

insignificant. It is also possible to obtain an estimate

of the collinearity of the variables x1 and x2 by using

the following relation [8]:

F = (N – m2)r1,22/[(m2 – 1)(1 – r1,22)] =

4.67 < F0.05cr(f1 = 1; f2 = 10) = 4.90. (21)

Inequality (21) also indicates that the explanatory

variables εnb0 and μ12 can be recognized as

independent. Consequently, the explanatory

variables x1 and x2, have a simultaneous effect on

the resultant variable, in a significant,

multidirectional and independent manner.

The regression coefficients (15) need to be

transformed again to reveal the comparative effect

on bioactivity of the explanatory variables:

B1* = B1Sx1/SA = -0.656, B2* = B2Sx2/SA = 0.428.

(22)

Thus, in contrast to the results (15), the standardized

coefficients B1* and B2* do not differ very

significantly in absolute value. Using the

approximate relation (13), we determine the share

contribution of each variable to the explanation of

the variability of bioactivity:

Rappr2 = B1*∙rx1,A + B2*∙rx2,A = 0.562 + 0.377 = 0.939.

(23)

The largest contribution to (23) is made by the

explanatory variable εnb0 (56.2%). The approximate

coefficient of determination Rappr2 = 0.939 is very

close to the coefficient of determination (16): R22 =

0.937.

In order to compare the coefficients of

determination of the two models with different

numbers of explanatory factors m, their adjusted R*2

values must be calculated (14). For regression (8),

the adjusted coefficient of determination is R1*2 =

0.90 for m3 = 3. However, for multiple regression

(16) R2*2 = 0.933; m2 = 2. The difference in the

coefficients of determination |R1*2 – R2*2| determines

the measure of additional explanation for the

variation of the resultant variable, by including

another explanatory variable in the regression.

Next, we use the following relation, which has the

Fisher F-distribution:

F = |R1*2 – R2*2|∙(N – m1 – 1)/(m1 – m2)/(1 – R1*2).

(24)

If F > F0.05cr(f1 = m1 – m2; f2 = N – m1 – 1), then the

additional explanatory variable must be retained in

the regression equation. In this case there is an

inverse inequality of F = 2.75 < F0.05cr(f1 = 1; f2 = 8)

= 5.32. That is, the additional explanatory variable

x3 does not improve the regression. The purpose of

the regression equation is not only to describe the

experiment satisfactorily. The regression equation

should indicate the physical phenomena associated

with the variability of bioactivity.

Thus, independent and simultaneously acting

explanatory molecular variables εnb0 and μ12 are

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

122

Volume 21, 2022

associated with the toxicity of substituted

chlorobenzenes. Model (15) makes it possible to

make some assumptions about the properties of the

biophase region with which an exogenous molecule

can interact. In accordance with the regression

equation (15), biophase molecules must have a

significant dipole moment or charge. In addition, the

energy of the highest occupied molecular orbital

εhocc of the biophase should be close to the energy

εnb0 of the exogenous molecule. If it is possible to

briefly describe the initial information, then there is

confidence that some objective regularity has been

revealed that exists in the structure of the feature

space, allowing this reduction to be carried out [18].

The correlation between the experimental toxicity

values of substituted chlorobenzenes and theoretical

values is shown in Fig. 1. The energy of

stabilization of the molecular complex is the higher,

the greater the energy of the donor-acceptor

complex.

Table 1 also lists the Z and H molecular features for

chlorine-substituted compounds. Z is the average

number of electrons in the outer shell of atoms in a

molecule: Z = ΣiniZi/N [36,37]. Here ni is the

number of atoms of the i-th sort with the number of

electrons Zi on the outer electron shell. The

summation is performed on all atoms in the

molecule; Σini = N is the total number of atoms. The

electronic factor Z is related to the pseudopotential

of the molecule [38].

Fig. 1. Scatterplot and regression line. Calculated

and experimental values of average lethal doses (A ≡

1000/LD50) of chlorobenzene derivatives (Table 1).

The regression line is given by the equation Amod =

a0 + a1Aexp, N = 12, a0 = 0.191 ± 0.168, a1 = 0.904 ±

0.080; R = 0.96 ± 0.03; R* = 0.97 > R0.05cr(N – 2) =

0.576, F = 159.2 > F0.05cr(f1 = 1; f2 = 10) = 4.96;

straightness index: K = (N∙(1 – R2))0.5 = 0.84 <

Kthr(threshold value) = 3.00 [35].

The information function H [39], for a discrete data

set, is quantified as follows: H = - Σjpjlog2pj. The

ratio pi = ni/N satisfies the following conditions: 0 ≤

pi ≤ 1, Σipi= 1. In which connection, pi = 0 means

the impossibility of the occurrence of the i-th event;

Σini = N; N is the number of atoms in the molecule.

The ratio nk/N determines the share holding of the

kth kind of atom in the molecule. For chemical

compounds from Table 1, it was found that the

molecular factor Z is closely related to the

polarizability of the α1 molecule: the linear

correlation coefficient is 0.93 ± 0.12. The statistical

significance of the correlation coefficient is

determined by the inequality:

t = |R|∙(N – 2)0.5/(1 – R2)0.5 =

12.36 >t0.05cr(N – 2) = 2.23. (25)

Inequality (25) uses a two-sided critical region for

the t-quantile. The Chaddock scale defines the

pairwise correlation coefficient as corresponding to

a “very close relationship” [40].

The relationship between the values of one-electron

energies εnb0 and the values of the molecular factor Z

was also verified. A very close linear relationship

was found (Fig. 2A) between these factors for

chlorine-substituted benzenes (chemical compound

numbers No = 1 – 6 in Table 1):

εnb0(Z)1 = a01 + a11∙Z, N1 = 6, R1 = -0.996 ± 0.004,

|R1*| = 0.998 > R0.05cr(N1 – 2) = 0.811; criterion of

significance of the correlation coefficient based on

the normalizing Fisher z-transform (Hotelling

corrections is taken into account [33]): uH = 2.98 >

u0.05(N) = z0.975∙(N – 1)-0.5 = 0.86; S1 = 0.040;

sufficient sample size to ensure the validity of the

correlation coefficient: N0.05min < 4; a01 = 1.23 ±

0.13, a11 = -0.75 ± 0.03, |t(a11)| = 21.7 > t(a01) =

9.46 > t0.05cr(N1 – 2) = 2.776; F = 469.5 > F0.05cr(f1 =

1;f2 = 4) = 7.71; straightness index: K = 0.22 < Kthr =

3.0. (26)

Because the corrected correlation coefficient |R1*| =

0.998, and the value of S1 = 0.040, then this

relationship of features is close to the functional

dependence.

Z and εnb statistics:

N1 = 6, Zav = 3.75 ± 0.21; 95% confidence interval

(3.20-4.30), Zmin = 3.00, Zmax = 4.50, SZ1 = 0.524,

τmax = 1.43 = τmin = 1.43 < τ0.05cr,2(N1) = 1.996 <

τ0.05cr,1(N1) = 2.184; Wilk-Shapiro normality test: W

= 0.960 > W0.05cr(N1) = 0.788, David-Hartley-

Pearson normality test: U10.05cr(N1) = 2.200 < U =

[(Zmax – Zmin)/SZ] = 2.863 < U20.05cr(N1) = 3.012, Nrepr

= 5; (27)

N1 = 6, εnbav = -1.56 ± 0.16; 95% confidence interval

(-1.97,-1.15), εnbmin = -2.097, εnbmax = -0.971, Sε =

0 1 2 3 4 5

0

1

2

3

4

5

À (experiment)

À (model)

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

123

Volume 21, 2022

0.392, τmin = 1.37 < τmax = 1.50 < τ0.05cr,2(N1) =

1.996< τ0.05cr,1(N1) = 2.184; Wilk-Shapiro normality

test: W = 0.975 > W0.05cr(N1) = 0.788, David-

Hartley-Pearson normality test: U10.05cr(N1) = 2.200

< U = [(εnbmax – εnbmin)/Sε] = 2.870 < U20.05cr(N1) =

3.012; Nrepr = 5. (28)

A B

Fig. 2. Scatterplots and regression lines. Relationship between the energy of the lowest free molecular orbital

Enb ≡ εnb0 of an exogenous molecule and the molecular factor Z. (A) Chloro-substituted benzenes. (B)

Nitrochlorobenzenes.

The sets Z and εnb are homogeneous and have a

normal distribution. A quantitative relationship

between these features was also found (Fig. 2B) for

chlorine-substituted nitrobenzenes (numbers of

chemical compounds Nos. = 8 - 12 in Table 1):

εnb0(Z)2 = a02 + a12Z, N2 = 5, R2 = - 0.89 ± 0.12,

|R2*| = 0.94 > R0.05cr(N2 – 2) = 0.8783; S2 = 0.166;

criterion of significance of the correlation

coefficient based on the normalizing Fisher z-

transform (Hotelling corrections is taken into

account): uH = 1.904 > u0.05(N) = z0.975∙(N – 1)-0.5 =

0.98; sufficient sample size to ensure the validity of

the correlation coefficient: N0.05min < 4; a02 = -1.37 ±

0.59, a12 = -0.47 ± 0.14, |t(a12)| = 3.32 > t0.05cr(N2 –

2) = 3.182; F = 11.04 > F0.05cr(f1 = 1; f2 = 3) =

10.13; straightness index: K = 1.02 < Kthr = 3.0. (29)

Z and εnb statistics:

N2 = 5, Zav = 4.11 ± 0.26; 95% confidence interval

(3.38-4.83), Zmin = 3.71, Zmax = 5.00, SZ2 = 0.584,

τmin = 0.682 < τmax = 1.527 < τ0.05cr,2(N2) = 1.869<

τ0.05cr,1(N2)= 2.080; Wilk-Shapiro normality test: W =

0.773 > W0.05cr(N2) = 0.762, David-Hartley-Pearson

normality test: U10.05cr(N2) = 2.200 < U = [(Zmax –

Zmin)/SZ] = 2.21 < U20.05cr(N2) = 3.222; Nrepr = 4;

N2 = 5, εnbav = -3.28 ± 0.14; 95% confidence interval

(-3.66,-2.30), εnbmin = -3.657, εnbmax = -2.987, Sε =

0.310, τmax = 0.95 < τmin = 1.22 < τmax = 1.527

τ0.05cr,2(N2) = 1.869 < τ0.05cr,1(N2) = 2.080; Wilk-

Shapiro normality test: W = 0.831 > W0.05cr(N2) =

0.762, David-Hartley-Pearson normality test:

U10.05cr(N2) = 2.200 ≈ U = [(εnbmax – εnbmin)/Sε] = 2.16

< U20.05cr(N2) = 3.012, Nrepr = 4. (30)

An approximate comparative estimate of the

regression coefficients a11 and a12 can be made

using the following relation [41]:

t = |a11 – a12|/[S12/(N1 – 1)/SZ12 + S22/(N2 – 1)/SZ22]0.5

= 1.918 < t0.05cr(N1 + N2 – 4) = 2.365. (31)

That is, estimates of regression coefficients differ

insignificantly, since t < tcr.

Dispersion interaction does not make a statistically

significant contribution to the explanation of the

bioresponse of a number of compounds from Table

1. However, there is a relationship between the

molecular feature Z and the value of the dispersion

contribution:

x3(Z) = a0 + a1Z, N = 12, R = 0.96 ± 0.03, R* = 0.97

> R0.05cr(N – 2) = 0.576; sufficient sample size to

ensure the validity of the correlation coefficient:

N0.05min < 5; criterion of significance of the

correlation coefficient based on the normalizing

Fisher z-transform (Hotelling corrections is taken

into account): uH = 1.942 > u0.05(N) = z0.975∙(N – 1)-0.5

= 0.591; a0 = -1.10 ± 0.95, a1 = 2.52 ± 0.24, t(a1) =

10.42 > t0.05cr(N – 2); standard error of the regression

estimate: 0.44; F = 108.5 > F0.05cr(f1 = 1;f2 = 10) =

4.96. (32)

Statistics of set Z:

3 3.5 4 4.5 5

2.5

2

1.5

1

0.5

Z, arb. units.

Enb, eV

3.5 4 4.5 5 5.5

3.8

3.6

3.4

3.2

3

2.8

Z, arb. units

Enb, eV

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

124

Volume 21, 2022

N = 12, Zav = 3.87 ± 0.16; 95% confidence interval:

3.52 - 4.22; Zmin = 3.00, Zmax = 5.00, SZ = 0.548, τmin

= 1.59 < τmax = 2.06 < τ0.05cr,2(N) = 2.387 <

τ0.05cr,1(N) = 2.523; Wilk-Shapiro normality test: W =

0.951 > W0.05cr(N) = 0.859, David-Hartley-Pearson

normality test: U10.05cr(N) = 2.800 < U = [(Zmax –

Zmin)/SZ] = 3.64 < U20.05cr(N) = 3.910; Nrepr = 10.

(33)

It follows that the sets of indices Z is homogeneous

and satisfies a normal distribution. The authors of

[42] present the additive contributions to the energy

of the pair interaction of a tetramethyluric acid

molecule with aromatic hydrocarbon molecules.

Including the magnitude of the dispersion

contribution is reported. Our check showed that in

this case, too, there is a statistically significant

relationship between the molecular factor Z and the

dispersion energy value of the pairwise interaction.

If there are chemical compounds capable of forming

hydrogen bonds, the energy contribution due to

hydrogen bonds must be taken into account in the

regression equations (8) and (15). The hydrogen

bond is essentially a quasi-chemical short-range

interaction of molecules. Therefore, the properties

of a hydrogen bond are difficult to describe using

the properties of isolated molecules. However, some

quantitative estimates can be made. Considering that

the terms of equation (8) are proportional to the

contributions to the binding energy, it can be

assumed that the contributions to the bioactivity

from these interactions are also approximately

equal.

The application of model (15) to the calculation of

the bioactivity of the 2,4-dichlorophenol molecule

leads to the following value of the resultant feature

A = 0.58 (εnb0 = -1.4eV, μ1 = 1.5D), which is

markedly lower than the experimental value 2.08.

However, it should be kept in mind that in an

isolated 2,4-dichlorophenol molecule there is an

intramolecular hydrogen bond between the hydroxyl

group proton and the chlorine atom in the ortho-

position. When a molecule enters a polar condensed

medium, the intramolecular hydrogen bond is

broken due to the electric field of the reaction. This

state of the molecule corresponds to a lower total

energy. In other words, the molecular state is

stabilized and the hydroxyl group has the

opportunity to take part in the formation of an

intermolecular hydrogen bond. In a polar dielectric

medium, with an increase in the dipole moment of a

molecule, the equilibrium of molecular forms with

and without an intramolecular hydrogen bond shifts

towards a molecular state with a large dipole

moment. That is, without the formation of an

intramolecular hydrogen bond. Such a situation has

indeed been observed experimentally. The

formation of an intermolecular hydrogen bond is

accompanied by an additional contribution to the

interaction energy. This in turn increases the

bioresponse A by about 1.0 (contribution to the total

interaction energy ≈ - 0.85εnb0). Therefore, the total

calculated value of bioactivity A will be

approximately equal to 1.58, which is close to the

experimental value. However, these remarks cannot

be applied to the 2,4,6-trichlorophenol molecule

(Table 1). This molecule contains a hydroxyl group.

However, the molecule is not involved in the

formation of intermolecular complexes through

hydrogen bonds. The fact is that the proton of the

hydroxyl group oscillates between two neighboring

chlorine atoms, which are characterized by

significant electronegativity. Oscillations are the

result of a proton tunneling through a potential

barrier. In this way, an intramolecular transition

from one equilibrium position to another is carried

out. Spectroscopic studies of 2,4,6-trichloro-,

2,4,5,6-tetrachloro- and pentachlorophenols confirm

the existence of intramolecular proton migration.

The application of the regression equation (8) to

predict the bioactivity of molecules is complicated

by the fact that the researcher is required to know

many molecular parameters. In particular,

knowledge of the energies of single-electron

molecular orbitals, which can be determined by

complicated and cumbersome quantum mechanical

calculations, is required. The results of these

calculations require professional analysis.

Fig.3. Scatterplot and regression line. The

regression line is approximated by the linear

equation: А(H)mod = b0 + b1H, N = 12; b0 = -5.25 ±

1.15, b1 = 4.02 ± 0.07, N0.05min = 5; R = 0.89 ± 0.07;

R* = 0.90 > R0.05cr(N – 2) = 0.576, S = 0.604, F =

37.6 > F0.05cr(f1 = 1; f2 = 10) = 4.96; straightness

index: K = 1.59 < Kthr = 3.0.

However, as the analysis showed, some rapid

assessment of the toxicity of chlorine-substituted

1.2 1.4 1.6 1.8 2 2.2

1

0

1

2

3

4

H, bits

Bioactivity

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

125

Volume 21, 2022

benzenes can be done using the information

function H (Table 1). Figure 3 shows a linear

relationship between the toxicity of chlorine-

substituted benzenes and the information function of

molecules. Using the regression equation in Fig.3,

we obtain the following estimate for the bioactivity

of 2,4-dichlorophenol (Aexp = 2.08), which was not

used in the original sample: Amod = 1.74 (H = 1.738

bits, Z = 3.692 arb. units.).

There is also a significant relationship between the

toxicity of chemical compounds and the explanatory

factor Z:

А(Z) = b0 + b1Z, N = 12, R = 0.62 ± 0.20, R* = 0.64

> R0.05cr(N – 2) = 0.576; sufficient sample size to

ensure the validity of the correlation coefficient:

N0.05min = 10; b0 = -3.82 ± 2.29, b1 = 1.43 ± 0.59,

t(b1) = 2.44 > t0.05cr(N – 2) = 2.228; standard error of

the regression estimate: SA = 1.044; criterion of

significance of the correlation coefficient based on

the normalizing Fisher z-transform (Hotelling

corrections is taken into account): uH = 0.697 >

u0.05(N) = z0.975∙(N – 1)-0.5 = 0.591; straightness

index: K = 2.7 < Kcr = 3.0. (34)

For 2,4-dichlorophenol, we obtain the following

toxicity estimate from equation (34): A = 1.46 at Z =

3.692 arb. units. The higher the Z and H values, the

higher the toxicity of the chemical compound. The

existence of such a trend for factor Z is also

indicated by the Abbe-Linnick criterion [27]:

q = 0.5∙Σi=1N-1(Zi+1 – Zi)2/Σi=1N(Zi – Zav)2 =

0.415 < q0.05cr(N) = 0.5636,

Q* = - (1 – q)∙[(2N + 1)/(2 – (1 – q)2)]0.5 =

-2.27 < u0.05 = -1.645. (35)

A similar trend takes place for the molecular

information factor H:

q = 0.231 < q0.05cr(N) = 0.5636,

Q* = -3.24 < u0.05 = - 1.645. (36)

Regression (15) indicates only an important initial

stage of the manifestation of its biological action by

a chemical compound, associated with the fixation

of the molecule.

3.2 Saturated and unsaturated chlorine-

containing compounds

In this part of the article, we will again use the

general equation (8) to interpret the narcotic effect

of a series of saturated and unsaturated chlorine-

containing compounds. Isotoxic concentrations (C

in units of mM/l) of vapours of compounds causing

lateral positioning of 50% of white mice are used as

the biological response (Table 2). We use the

following regression equation, which is similar in

form to regression (8):

Amod ≡ 100/C = B0 + B1x1 + B2x2 + B3x3,

N = 15, B0 = -11.88 ± 1.11, B1 = - 0.96 ± 0.35, B2 =

0.90 ± 0.13, B3 = 2.72 ± 0.27, R3 = 0.980 >

R0.05cr(m3; N – m3 – 1) = 0.703, R32 = 0.960, R3*2 =

0.949; m3 = 3; standard error of the regression

estimate: S3 = 0.905; |t(B0)| = 10.71 > t(B3) = 10.24

> t(B2) = 6.94 > t(B1) = 2.79 > t0.05cr(f = N – m3 –

1) = 2.20; F = 88.04 > F0.05cr(f1 = 3; f2 = 11) = 3.59;

Σ3 = 9.0072; AIC3 = - 0.1100, SC3 = 0.2121, SS3 =

0.2501. (37)

Bioactivity statistics:

A: N = 15, Aav = 5.27 ± 1.04; 95% confidence

interval: 3.04 - 7.49; Amin = 0.71, Amax = 13.33, SA

= 4.01, τmin = 1.14 < τmax = 2.01 < τ0.05cr,2(N) =

2.493 < τ0.05cr,1(N)= 2.617; Wilk-Shapiro normality

test: W = 0.881 = W0.05cr(N) = 0.881, David-Hartley-

Pearson normality test: U10.05cr(N) = 2.970 < U =

[(Amax – Amin)/SA] = 3.15 < U20.05cr(N) = 4.170; Nrepr

= 12.

Statistics of explanatory variables:

x1: N = 15, x1av = -1.19 ± 0.27; 95% confidence

interval: -1.77, -0.61; x1min = -2.75, x1max = 0.45, Sx1

= 1.04, τmax = 1.04 < τmin = 1.50 < τ0.05cr,2(N) =

2.493 < τ0.05cr,1(N) = 2.617; Wilk-Shapiro normality

test: W = 0.930 > W0.05cr(N) = 0.881, David-Hartley-

Pearson normality test: U10.05cr(N) = 2.970 < U =

[(x1max – x1min)/Sx1] = 3.08 < U20.05cr(N) = 4.170, Nrepr

= 12,

x2: N = 15, x2av = 3.59 ± 0.61; 95% confidence

interval: 2.29 - 4.89; x2min = 0, x2max = 7.95, Sx2 =

2.35, τmin = 1.53 < τmax = 1.86 < τ0.05cr,2(N) =

2.493 < τ0.05cr,1(N) = 2.617; Wilk-Shapiro normality

test: W = 0.957 > W0.05cr(N) = 0.881, David-Hartley-

Pearson normality test: U10.05cr(N) = 2.970 < U =

[(x2max – x2min)/Sx2] = 3.39 < U20.05cr(N) = 4.170;

Nrepr = 12,

x3: N = 15, x3av = 4.71 ± 0.30; 95% confidence

interval: 4.07 - 5.34; x3min = 3.308, x3max = 7.353,

Sx3 = 1.149, τmin = 1.22 < τmax = 2.31 <

τ0.05cr,2(N) = 2.493 < τ0.05cr,1(N) = 2.617; Wilk-

Shapiro normality test: W = 0.923 > W0.05cr(N) =

0.881, David-Hartley-Pearson normality test:

U10.05cr(N) = 2.970 < U = [(x3max – x3min)/Sx3] = 3.52

< U20.05cr(N) = 4.170; Nrepr = 12. (38)

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

126

Volume 21, 2022

Thus, the populations x1, x2, x3 and A are

homogeneous and normally distributed.

For regression (37), the t-values of all coefficients

|t(Bi)| > t0.05cr(N). Consequently, the coefficients

characterize the significant effect of each of the

intermolecular interaction contributions on the

toxicity of chemical compounds. To determine the

comparative influence of individual contributions of

intermolecular interactions, we turn to standardized

regression coefficients. We will use relations (22)

for this purpose:

B1* = B1Sx1/SA = -0.251, B2* = B2Sx2/SA = 0.525,

B3* = B3Sx3/SA = 0.778. (39)

Let us determine the contribution of each variable to

the variability of bioactivity. The approximate

value of the multiple coefficient of determination is

determined by the relation (13):

Rappr2 = B1*∙rx1,A + B2*∙rx2,A + B3*∙rx3,A =

0.098 + 0.179 + 0.686 = 0.963. (40)

Here, the pair correlation coefficients are rx1,A = -

0.392, rx2,A = 0.337 and rx3,A = 0.880. The

approximate multiple coefficient of determination

(40) practically coincides with the value of the

coefficient of determination R32 = 0.960 (37). Given

the values of the standardized regression

coefficients (39), it can be noted that the

explanatory variables do not equally affect the

variability of the bioactivity. In contrast to the series

of chlorobenzenes (Table 1), for which the

contribution from the variable x1 = εnb0 is maximum,

for a number of chemical compounds from Table 2

this contribution to regression (37) is minimal. At

the same time, the maximum contribution to the

variability of bioactivity is made by pair dispersion

interactions x3 = α1I1/(I1 + 10). For chlorobenzenes,

the sequence of influence on the bioactivity of

intermolecular interactions will be as follows (13):

x1(53.5%) > x2(32.1%) > x3(7.5%). Whereas for a

number of compounds from Table 2 the hierarchy of

influence will be the opposite: x1(9.8%) < x2(17.9%)

< x3(68.6%). The parentheses indicate the

percentage contributions of the explanatory

variables. Note that the signs of the coefficients of

Bi* (37) remain the same as for the regression

equation (10). Consequently, the direction of

influence of the explanatory variables on the

bioactivity of the molecules does not change. Thus,

the molecular factors taken into account by the

model explain 96.0% of the variation in the

bioresponse (R32 = 0.960) and only 4.0% remain

unexplained.

The hierarchy of values of the regression

coefficients Bi* makes it possible to indicate the

relative importance of the intermolecular

contributions taken into account in the regression.

The maximum contribution to the regression

equation is made by the explanatory variables that

determine the pair dispersion interaction. The

influence of acceptor interactions (~ εnb0) can be

considered as corrections to the dispersion and

dipole-dipole contributions. It seems that the

molecular region of the biophase with which the

molecule interacts has the highest filled molecular

orbital energy level, which lies below the εnb0 level

on the energy scale. That is, such an arrangement of

energy levels does not favor the transfer of an

electron to a free one-electron level εnb0.

Testing for collinearity between explanatory

variables resulted in the following values for the

pairwise correlation coefficients between

explanatory variables: r1,2 = 0.541, r1,3 = -0.545 и

r2,3= 0.064. To quantitatively check the presence of

collinearity between explanatory variables, we use

the Farrar-Glauber test (19):

χ2 = - (N – 1 – (2m + 5)/6)∙ln(det|ri,j|) = 12.16 >

χ0.052,cr (f = m(m – 1)/2) = 7.82; i = 1,2,3; j = 1,2,3.

(41)

Since χ2 > χ2,cr, it is necessary to reject the null

hypothesis about the absence of collinearity between

the explanatory variable at a significance level of α

= 0.05. To determine which explanatory variable

generates the greatest interdependence between

variables, the following relationship is used [8]:

ti,k = ri,k(N – m)0.5/(1 – ri,k2)0.5 , (42)

which has a t-distribution with f = N – m degrees of

freedom. Using relation (42), we obtain the

following sequence of inequalities: |t1,3| = 2.251 >

t1,2 = 2.228 > t0.05cr(f = N – m3) = 2.179 > |t2,3| =

0.222. From these inequalities it follows that the

variable x1 generates the interdependence of

features.

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

127

Volume 21, 2022

Table 2

Physico-chemical parameters of chlorine-containing compounds and isotoxic concentrations (C, mM/l) of the

vapors of these compounds, causing the lateral position of 50% of white mice.

№

Chemical compound

Z,

arb.

units

H,

bits

d20, g/ml [43]

n20 [43]

εnb0, eV

I1*), eV

μ1*), D,

α1**), 1024cm3

A exp., 100/C,

[44]

Amod,, Eq.(37)

1

Ethyl chloride

2.50

1.299

0.8978

1.3676

0.151

10.70

2.0 [14]

6.40

0.71

0.55

2

Propyl chloride

2.36

1.241

0.8909

1.3879

0.445

10.44

1.8 [14]

8.24

1.23

2.03

3

Vinyl chloride

3.00

1.460

0.9999

1.4046

-0.522

9.82

1.44[14]

7.83[15]

1.56

1.02

4

1,1-Dichlorovinyl

4.00

1.586

1.2180

1.4249

-1.216

9.59

1.13

8.07

2.50

1.17

5

1,2-Dichlorovinyl

4.00

1.586

1.2837

1.4490

-1.214

9.15

1.77

7.78[15]

2.50

2.20

6

1,1-Dichloroethane

3.25

1.500

1.1757

1.4164

-1.206

10.60

1.80[14]

8.38

3.08

3.90

7

Chloromethylene

2.80

1.372

1.3255

1.4242

-1.215

10.79

2.40[14]

6.48[15]

3.08

3.59

8

Trichlorovinyl

3.45

1.539

1.4642

1.4773

-2.276

9.48

1.01

10.06

4.00

4.53

9

Chloroform

4.86

1.449

1.4832

1.4459

-2.467

11.00

1.51

8.23

5.00

4.25

10

Tetrachlorovinyl

6.00

0.919

1.6227

1.5053

-2.623

9.66

0

12.03

5.00

6.70

11

1,2-Dichloroethane

3.25

1.500

1.2531

1.4476

-0.089

10.92

2.43

8.45

5.71

5.49

12

1,2-Dichloropropane

2.91

1.436

1.1559

1.4394

0.099

10.60

2.82

10.20

9.52

9.42

13

1,1,2-Trichloroethane

4.00

1.562

1.4714

1.4940

-1.264

10.89

2.72

10.28

10.00

10.5

14

1,1,2,2-Tetrachloroethane

4.75

1.500

1.5953

1.4940

-1.680

10.89

2.17

12.15

11.76

11.2

15

Pentachloroethane

5.50

1.299

1.6796

1.5025

-2.753

10.88

1.37

14.11

13.33

12.4

*) Dipole moments and ionization potentials of the molecules are calculated by the quantum-chemical method MINDO/3.

**) Polarizabilities of molecules are calculated using the Clausius-Mossotti formula: α = (n202 - 1)3M/(n202 + 2)/(4πd20NA);

d20 and n202 are the density and refractive index at 200C, respectively.

To eliminate or reduce the collinearity of the

explanatory variables, perform a linear

transformation. For example, for the variable x2 : x2*

= x1 – x2. We write multiple regression (37) as

follows:

Amod ≡ 100/C = B0 + B1x1 + B2x2* + B3x3,

N = 15, B0 = -11.88 ± 1.11, B1 = -0.064 ± 0.287, B2

= - 0.897 ± 0.129, B3 = 2.716 ± 0.265, R3 = 0.980

> R0.05cr(m3 = 3; ν = N – m3 – 1) = 0.703, R32 =

0.960, R3*2 = 0.954; m3 = 3; standard error of the

regression estimate: SA = 0.905; |t(B0)| = 10.71 >

t(B3) = 10.24 > t(B2) = 6.94 > t0.05cr(f = N – m3 – 1)

> |t(B1)| = 0.224; F = 88.06 > F0.05cr(f1 = 3;f2 = 11) =

3.59 = 3.59; Σ3 = 9.007, AIC3 = -0.1100, SC3 =

0.2121, SS3 = 0.2501; B1* = -0.0168, B2* = -0.4448,

B3* = 0.7783. (43)

The approximate coefficient of determination Rappr2

practically coincides with the coefficient of

determination R32= 0.960 (43):

Rappr2 = B1*∙rx1,A + B2*∙rx*2,A + B3*∙rx3,A =

0.007 + 0.269 + 0.685 = 0.961. (44)

The regression coefficient B1 (43) is statistically

insignificant at the 95% confidence level.

Correlation coefficients were also determined

between the explanatory variables x1, x2* and x3: r1,2*

= 0.112, r1,3 = - 0.545 and r2*,3= - 0.207. Between

variables x1 and x2* there was a significant decrease

in the correlation coefficient (compare with r1,2 =

0.541). From the Farrar-Glauber relation (41) we

now obtain the following inequality: χ2 = 5.06 <

χ0.052,cr(f = 3) = 7.82. Thus, the null hypothesis that

there is no significant multicollinearity between the

explanatory variables can be accepted.

Statistics of the population of the explanatory

variable x2*:

N = 15, x2*av = - 4.78 ± 0.51; 95% confidence

interval: (-5.88, - 3.68); x2*min = -8.66, x2*max = -

2.49, Sx*2 = 1.989, τmax = 1.151 < τmin = 1.951 <

τ0.05cr,2(N) = 2.493 < τ0.05cr,1(N) = 2.617; Wilk-

Shapiro normality test: W = 0.919 > W0.05cr(N) =

0.881, David-Hartley-Pearson normality test:

U10.05cr(N) = 2.970 < U = [(x2*max – x2*min)/Sx*2] =

3.10 < U20.05cr(N) = 4.170. (45)

Therefore, the set of elements x2* is homogeneous

and normally distributed. The influence of the

explanatory factor x1 = εnb0 on the variability of

bioactivity is markedly smaller compared to the

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

128

Volume 21, 2022

contributions from variables x2 (or x2*) and x3.

Therefore, we will perform the following

comparative analysis. Write the following reduced

regression:

A ≡ 100/C = B0 + B2x2 + B3x3. (46)

The regression (46) statistics will be as follows:

N = 15, m4 = 2, R4 = 0.965 > R0.05cr(m4; N – m4 – 1)

= 0.627, R42 = 0.932, R4*2 = 0.921; standard error

of the regression estimate: S4 = 1.131; B0 = -12.063

± 1.384, B2 = 0.681 ± 0.129, B3 = 3.164 ± 0.264;

|t(B3)| = 12.00 > |t(B0)| = 8.72 > t(B2) = 5.28 >

t0.05cr(f = N – m4 – 1) = 2.18, F = 82.04 > F0.05cr(f1

= 2;f2 = 12) = 3.88; Σ4 = 15.354, AIC4 = 0.2900,

SC4 = 0.5649, SS4 = 0.3014. (47)

There is no collinearity between variables x2 and x3

(r2,3 = 0.064). The multiple coefficient of sample

correlation R4 significantly exceeds the tabular value

of the correlation coefficient R0.05cr(m4; N – m4 – 1).

Thus, 93.2% of the total variance is due to the

variability of the molecular factors x2 and x3. The

uncertainty factor is 6.8%. However, all three

comparative information tests AIC4, SC4, SS4

indicate that the quality of the regression (46) is

markedly reduced compared to the regression (37).

The standardized regression coefficients (46) are:

B2* = 0.399, B3* = 0.907. (48)

Adjusted coefficients of determination are R3*2 =

0.95 for regression (37) and R4*2 = 0.921 for reduced

regression (46), respectively. Using the standardized

regression coefficients (48) an approximate

coefficient of determination can be determined:

Rappr2 = B2*∙ rx2,A + B3*∙ rx3,A =

0.14 + 0.80 = 0.94. (49)

This value of the coefficient of determination is very

close to the value of R42 = 0.932 (47).

The significance of the contribution of x1 to the

regression equation (37) can be checked with the

following statistics:

F = (|R3*2 – R4*2|)(N – m3 – 1)/(m3 – m4)/(1 – R3*2) =

6.04 > F0.05cr(f1 = m3 – m4; f2 = N – m3 – 1) = 4.84.

(50)

Since F > Fcr the additional explanatory variable εnb0

is significant at the 95% confidence level. Let us

also check whether the decrease in the variance of

equation (37) is the result of an increase in the

number of connections compared to regression (46).

For normally distributed sets, the comparison of the

two variances of equations (37) and (46) is done

using a statistic that has an F-distribution:

F = S42/S32 = 1.56 <

F0.05cr(f1 = N – m4 – 1; f2 = N – m3 – 1) = 2.79. (51)

Therefore, at the 95% confidence level, the decrease

in the variance of the regression equation (37) is not

due to an increase in the number of explanatory

variables. Thus, variable x1 must be retained in the

regression equation. All three AIC, SC and SS

information tests do not contradict inequality (50).

Preference is given to the model for which the

information test has the least value. The test

comparison is only valid for models built for

samples containing the same number of

observations. A comparison of the quality criteria of

regression equations (10), (16), (37) and (46) is

presented in Table 3.

Table 3

Quality criteria for linear regression equations

i

m

N

Σi

AICi

SCi

SSi

Equation

1

3

12

1.1586

-1.8377

-1,5093

0.1196

(10)

2

12

1.2294

-1.9450

-1.6572

0.1109

(16)

3

15

9.0072

-0.1100

0.2121

0.2501

(37)

4

2

15

15.354

0.2900

0.5649

0.3014

(46)

As expected, the quality criteria of the regressions

correlate with each other: RAIC-SC = 0.9997, RAIC-SS =

0.997 and RSC-SS = 0.995. From the inequalities for

the t-values of the regression coefficients (38), one

can indicate the following sequence of

intermolecular interactions: dispersion interaction

(short - range) > dipole - dipole, induction and

polarization interactions > interaction associated

with electron transfer. Each of the contributions of

this sequence has a very specific physical meaning.

This, in turn, makes it possible to associate

interactions with certain physical characteristics of

biophase molecules. Highlighting interactions is

useful for determining the presumed characteristic

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

129

Volume 21, 2022

molecular properties of the local centers with which

a bioactive molecule interacts.

The fact that the main contribution to the regression

equation comes from the dispersion interaction

allows us to make some assumptions about the

molecular properties of the biophase with which the

bioactive molecule forms a complex. According to

the definition (6) the interaction of molecules of a

number of chlorinated chemical compounds (Table

2) is most likely to take place with the biophase

region, which is characterized by high values of

electronic polarizability (α2) and has a high first

ionization potential (I2). It is also noteworthy that

for models (15) and (46) the dipole interaction turns

out to be significant. This result indicates the

importance of mutual orientations between bioactive

molecules and biophase molecules.

Figure 4 shows the correlation between the observed

bioactivity values of molecules and the calculated

bioactivities of chemical compounds using equation

(37).

A relationship was also found between the

dispersion contribution Edisp ~ x3 and the molecular

factor Z of chlorine-substituted hydrocarbons

presented in Table 2:

x3(Z) = a0 + a1Z, N = 15, R = 0.71 ± 0.14, R* =

0.73 > R0.05cr(N – 2) = 0.514; sample size sufficient

for the reliability of the correlation coefficient

N0.05min = 7; correlation coefficient significance test

based on the Fisher normalizing z-transform

(Hotelling corrections is taken into account): uH =

0.87 > u0.05(N) = z0.975∙(N – 1)-0.5 = 0.523; a0 = 1.91

± 0.81, a1 = 0.74 ± 0.21, t(a1) = 3.58 > t(a0) = 2.36

> t0.05cr(N – 2) = 2.16; standard error of the

regression estimate: 0.846; F = 12.85 > F0.05cr(f1 =

1; f2 = 13) = 4.70; straightness index: K = 2.73 < Kthr

= 3.00. (52)

Fig.4. Scatter plot and regression line. The

regression line is defined by the equation Amod = a0

+ a1Aexp; N = 15, a0 = 0.212 ± 0.355, a1 = 0.960 ±

0.054; R = 0.98 ± 0.01, R* = 0.982 > R0.05cr(N – 2) =

0.514; S = 0.81; F = 313.6 > F0.05cr(f1 = 1; f2 = 13) =

4.7; K = 0.77 < Kthr = 3.0.

Statistics of explanatory variable Z:

Z: N = 15, Zav = 3.78 ± 0.28; 95% confidence

interval: 3.17 - 4.38; Zmin = 2.36, Zmax = 6.00, SZ =

1.10, τmin = 1.29 < τmax = 2.03 < τ0.05cr,2(N) =

2.493 < τ0.05cr,1(N) = 2.617; Wilk-Shapiro normality

test: W = 0.934 > W0.05cr(N) = 0.881, David-Hartley-

Pearson normality test: U10.05cr(N) = 2.970 < U =

[(Zmax – Zmin )/SZ] = 3.31 < U20.05cr(N) = 4.170; Nrepr

= 12. (53)

The sets of elements x3 (39) and Z (53) satisfy the

homogeneity and normality conditions at the 95%

confidence level.

Fig.5. Scatter plot and regression line. The

regression line is defined by equation (54).

For the chemical compounds from Table 2 there is

also a significant relationship between the molecular

factor Z and the energy εnb0 (Fig. 5):

εnb0(Z) = a0 + a1Z, N = 15, R = -0.85 ± 0.08, |R*| =

0.85 > R0.05cr(N – 2) = 0.514; sample size sufficient

for the reliability of the correlation coefficient:

N0.05min = 5; correlation coefficient significance test

based on the Fisher normalizing z-transform

(Hotelling corrections is taken into account): uH =

1.179 > u0.05(N) = z0.975∙(N – 1)-0.5 = 0.523; standard

error of the regression estimate: 0.575; a0 = 1.87 ±

0.55, a1 = – 0.81 ± 0.14, |t(a1)| = 5.79 > t(a0) =

3.41 > t0.05cr(N – 2) = 2.16; F = 33.50 > F0.05cr(f1 = 1;

f2 = 13) = 4.70; straightness index: K = 2.04 < Kthr =

3.00. (54)

The statistical significance of the correlation

coefficient is characterized by the inequality (26): t

= 4.35 > t0.05cr(N – 2) = 2.16. According to the

Chaddock scale, the correlation coefficient is in the

range of 0.7 – 0.9, which is characterized as “close

relationship”.For the chemical compounds presented

in Table 2, a relationship was found between

bioactivity and the molecular factor Z. There is a

statistically significant trend between the Z value

and the bioactivity of chemical compounds. Indeed,

0 5 10 15

0

5

10

15

A (experiment)

A (model)

2 3 4 5 6

3

2

1

0

1

Z, arb. units

E(nb), eV

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

130

Volume 21, 2022

using the Abbe-Linnick test (35) we obtain the

following inequality:

q = 0.473 < q0.05cr(N = 15) = 0.6027,

Q* = - 2.24 < u0.05 = - 1.645. (55)

Here Zav = 3.78 is the arithmetic mean of the set Zi.

Since inequalities (55) are satisfied, the null

hypothesis about the absence of a trend is rejected.

The alternative hypothesis of a trend is accepted.

The following significant linear relationship of

bioactivity with factor Z was also obtained:

А(Z) = d0 + d1Z, N = 15, R = 0.54 ± 0.20, R* = 0.56

> R0.05cr(N – 2) = 0.514; sample size sufficient for

the reliability of the correlation coefficient N0.05min =

13; correlation coefficient significance test based on

the Fisher normalizing z-transform (Hotelling

corrections is taken into account): uH = 0.592 >

u0.05(N) = z0.975∙(N – 1)-0.5 = 0.523; d0 = -2.23 ±

3.35, d1 = 1.99 ± 0.95, t(d1) = 2.33 > t0.05cr(N – 2) =

2.160; standard error of the regression estimate: SA

= 3.50; F = 5.42 > F0.05cr(f1 = 1; f2 = 13) = 4.70.

(56)

It should be noted that regression (56) does not

make it possible to distinguish between cis- and

trans-isomers. The Z factor correlates with the

electronic polarizability of the molecule α1 (R = 0.72

> R0.05cr(N – 2) = 0.514; t = 3.14 > t0.05cr(N – 2) =

2.160). There is also a relationship (R = 0.71; t =

3.07 > t0.05cr(N – 2)) with the value of α1I1/(I1 + 10).

There is also a relationship between the Z factor and

the MO energy εnb0 (|R| = 0.85 > R0.05cr(N – 2) =

0.514; t = 4.35 > t0.05cr(N – 2) = 2.160) (Fig. 5). For

chemical compounds (Table 2) having the general

formula C2HxCly (sample volume N = 11) there is a

very close linear relationship between the factor Z

and the value of the one-electron MO energy εnb0: R

= - 0.95, |R*| = 0.96 > R0.05cr(N – 2) = 0.602; t =

5.18 > t0.05cr(N – 2) = 2.262. Note that the energy

εnb0 of the molecule actually characterizes the

affinity of the molecule for the electron.

When analyzing the data in Table 2, of the two

possible structures of the 1,2-dichlorvinyl molecule,

the cis-isomer structure corresponding to the lower

total energy state of the molecule in a polar medium

was chosen (assuming κs = 80 for the static

dielectric permittivity of the polar medium and n32 =

1.777 for the refractive index). The quantitative

determination of the difference in the total energies

of a molecule in cis- and trans-configurations is

associated with the calculation of the difference

between two large quantities, the accuracy of which

depends significantly on the accuracy of the

quantum-chemical method used. Nevertheless, some

approximate quantitative estimates can be made if

the same method for calculating the electronic

structure of molecules is used in both cases. It is

assumed that possible inaccuracies of the quantum-

chemical method can be compensated for. The total

electronic energies of the trans-isomer and cis-

isomer calculated using the CNDO/2 method are

Etrans = - 1295.97 eV and Ecis = - 1295.95 eV,

respectively. However, in a polar dielectric medium,

the energy of the dipole cis-isomer, compared to the

nondipole trans-isomer, decreases on the energy

scale by

ER = - (κs – 1)(n32 + 2)μ12/[3a3(2κs + n32)] = - 08 eV.

(57)

Here it is accepted: μ1 = 1.77D; a = 2.5 Å is the

effective size of the molecule. This energy value is

noticeably higher in absolute value of the thermal

energy of the translational motion of a molecule at

room temperature (thermal energy ≈ 0.02 eV). The

difference in the total electronic energies of the

molecules is practically compensated. Then,

according to Boltzmann's statistics, one can estimate

the number of molecules in cis- and trans-

configurations in a condensed dielectric medium as

follows:

Ncis= Ntransexp[(Etrans – Ecis – ER)/kBT] = 13Ntrans .

(58)

Consequently, at room temperature, there are more

than ten molecules in the cis-configuration per

molecule in the trans-configuration. That is, most of

the molecules in a polar dielectric medium will

preferably be in the cis-configuration.

Comparing the regressions (15) and (46), we can

make some assumptions about the selective nature

of the biological action at the molecular level of the

analyzed chlorine-containing chemical compounds.

In the first case (Table 1), chemical chlorinated

compounds are most likely to interact with those

areas in the body characterized by strong donor

properties. In the second case (Table 2), the active

sites of the biophase are more prone to the

formation of molecular associates, which are

stabilized due to dispersion and dipole interactions.

This is typical for the active centers of the biophase,

which have high polarization properties (large

values of polarizability α2). Since short-range

interactions are important for equations (15) and

(46), then for the manifestation of the biological

activity of chlorine-containing chemical

compounds, the molecules must approach the active

centers of the biosubstrate at close distances less

than 5Ǻ.

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

131

Volume 21, 2022

4 Conclusion

Thus, the studied variability in the toxicity of

chlorinated hydrocarbons (Tables 1 and 2) is

probably due to the accumulation (at least in the

initial stages of action) of exogenous chemical

compounds in the body's active centers. The change

in toxicity can be associated with the electronic

structure of exogenous molecules, which determines

their ability to form bound molecular complexes due

to various types of intermolecular interactions. In

the case when the donor electron level εdon lies

higher on the energy scale of the acceptor level εacc,

a real electron transfer from the impurity molecule

to the biophase molecule is possible. In this case, in

the local region of the biophase, through relaxation

mechanisms, energy is radiationless released,

approximately equal to the difference εdon - εacc. This

energy can be used to destroy the equilibrium

structure of the biophase.

Multiple regression (8), (15), (37) reflect the

existence of objective relationships between the

bioresponse and a variety of molecular factors that

characterize the interaction of molecules in a

condensed medium. The general structure of the

response function (8) with a clear physical meaning

of the explanatory variables is able to cover various

aspects of the manifestation of the toxicity

properties of the analyzed chlorine-containing

chemical compounds of two different classes.

Additive molecular features x1, x2 and x3 are

intensive indicators that determine the cause-and-

effect relationships of bioactivity - the molecular

structure of chemical compounds. At the same time,

the models statistically significantly reflect the

relationship between individual molecular factors

and the biological response of the body. It is

important to note that the relatively simple

mathematical formulas derived from rigorous

theoretical concepts made it possible to obtain a

statistically significant relationship between

bioresponse and the molecular parameters of

chemical compounds. Such a relationship is difficult

to establish by the usual direct deduction from the

general to the particular.

References:

[1] Zahradnik R., Arch. Int. Pharmacodyn.,

Vol.135, 1962, p.311.

[2] Zahradnik R., Experientia, Vol.18, 1962, p.534.

[3] Boček K., Kopecky J., Krivucova M.,

Experientia, Vol.20, 1964, p.657.

[4] Kopecky J., Boček K., Experientia, Vol.23,

1967, p.125.

[5] Hansch C., On the Use of Quantitative

Structure-Activity Relationships in Drug Design

(Review). Chem.-Pharm. Journal, Vol.14,

No.10, 1980, pp.15-30. (in Russian).

[6] Hansch C., Leo A., Substituent Constant for

Correlation Analysis in Chemistry and Biology.

John & Sons, New York, Chichester, Brisbane,

Toronto, 1979.

[7] Aibinder N. E., Bezdvorny V.N., Krasovitskaya

M. L., In: Study of Biological Action of New

Products of Organic Synthesis and Natural

Compounds. Perm. 91 p., 1981, (in Russian).

[8] Förster E., Rönz B., Metohden der

Korrelations- und Regressionsanalyse, Verlag

Die Wirtschaft Berlin, 290 p., 1979.

[9] Likeś J., Laga J., Zăkladnŷ. Statistice Tabulky,

Praha, 356 p., 1978.

[10] Kubini H., In: Biological Activity and Chemical

Structure, Amsterdam, 1977.

[11] Golubev A.A., Lublina E.I., Tolokontsev N.A.,

Quantitative Toxicology, Leningrad, 1973 (in

Russian).

[12] Mukhomorov V.K., Frumin G.T., Quantitative

Ratios of Bioactivity - Electronic

Characteristics of Halocarbons of the Aliphatic

Series, Chem.-Pharm. Journal, Vol.16, No.10,

1982, pp.70-74 (in Russian).

[13] Mukhomorov V.K., Structure-Activity

Relationships. Intermolecular Interactions and

Toxicity of Compounds. Toxicology Letters,

Vol.88, Issue S1, 1996, p.87.

[14] Osipov V.A., Minkin V.M., Reference Book on

Dipole Moments, Мoscow, 1965 (in Russian).

[15] Vereshchagin A.N., Polarizability of

Molecules, Moscow, 1980, (in Russian).

[16] Frumin G.T., Mukhomorov V.K., Comparative

Analysis of Statistical Distributions as the Basis

of Bioactivity-Structure Models, Chem.-Pharm.

Journal., Vol.16, No.10, 1982, pp,70-74, (in

Russian).

[17] Klopman M., Simonetta M., Fujimoto H.,

Reactivity of Molecules and Reaction

Pathways, Moscow, 1977 (in Russian).

[18] Shchembelov G.A., Ustinyuk V.M., Mamaev

V.M., Ischenko V.M., Gloriozov I.P., Luzhkov

V.B., Orlov V.V., Simkin V.Y., Pupyshev V.I.,

Burmistrov V.N., (1980). Quantum-Chemical

Methods of Molecule Calculation. Edited by

V.M. Ustynyuk, Moscow, Publishing House

Chemistry, 1980 (in Russian).

[19] Sutton L.E., Tables of Interatomic Distances,

Configuration in Molecules, London, 1958,

1965.

[20] Frölich H., Theory of Dielectrics, Oxford,

1958.

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

132

Volume 21, 2022

[21] Onsager L., Electric Moments of Molecules in

Liquids, J. Am. Chem. Phys., Vol.58, No. 8,

1936, p.1486-1493.

[22] Bakhshiev N.G., Spectroscopy of

Intermolecular Interactions, Leningrad, 1972

(in Russian).

[23] Willcams D., Schaad L., Murrel J., Deviations

from Pairwise Additivity in Intermolecular

Potentials, J. Chem. Phys., Vol.47, No 12,

1967, pp.4916-4923.

[24] Kestner N., Sinanoglu O., Effective

Intermolecular Pair Potentials in Nonpolar

Media, J. Chem. Phys., Vol.38, No.7, 1963,

pp.1730-1740.

[25] Gurevich L.V., Karachevtsev G.V., Kondratyev

V.N., Lebedev Yu. A., Medvedev V.A.,

Potapov V.K., Khodeev Yu. S., Energies of

Breaking Chemical Bonds. Ionization

Potentials and Electron Affinity, Moscow,

Nauka, 1974 (in Russian).

[26] Bazilevsky M.V., Method of Molecular Orbits.

Reactivity of Organic Molecules, Moscow,

Publishing House Chemistry, 1969 (in

Russian).

[27] Kobzar A.I., Applied Mathematical Statistics.

For Engineers and Scientists, FizMatLit.,

Moscow, 2006 (in Russian).

[28] Bolshev L.N., Smirnov N.V., Tables of

Mathematical Statistics, Moscow, Nauka, 1983

(in Russian).

[29] Akaike Hirotogu, A New Look at the Statistical

Model Identification, IEEE Transactions on

Automatic Control, Vol.19, No.6, 1974, pp.

716-723.

[30] Schwarz G., (1978). Estimating the Dimension

of Model, The Annals of Statistics, Vol.6, No.2,

1978, pp. 461-464.

[31] Dmitriev E.A., Mathematical Statistics in Soil

Science, Moscow State University Press, 1995.

[32] Draper N.R., Smith H., Applied Regression

Analysis, John Wiley & Sons, New York, 1996.

[33] Sachs L., Statistische Auswertungsmetoden,

Springer Verlag, Berlin, New York, 1972.

[34] Farrar D.E., Glauber R.R., Multicollinearity in

Regression Analysis: The Problem Revisited,

In: The Review of Economics and Statistics.

Vol.49, No.1, 2018, pp. 92-107.

[35] Zaitsev G.N., Mathematics in Experimental

Botany, Moscow, 1990, (in Russian).

[36] Mukhomorov V.K., Statistical Asptcts of the

Interrelation, Biomedical Statistics and

Informatics, Vol.1, No.1, 2016, pp. 24-34.

[37] Mukhomorov V.K., Entropy Approach to the

Study Biological Activity Chemical

Compounds, Advances in Biological

Chemistry, Vol.1, No.1, 2011, pp. 1-5.

[38] Veljkovič V., Lalovič D., General Model

Pseudopotential for Positive Ions, Phys. Lett. A,

Vol.45, No1, 1973, pp. 59-62.

[39] Quastler H., Information Theory in Biology,

Moscow, IL, 1960, (in Russian).

[40] Chaddock R.E., Principles and Methods of

Statistics, Boston, New York, 1925.

[41] Johnson N.L., Leone F.C., Statistics and

Experimental Design, Vol.1, Second Edition,

John Wiley & Sons, New York, London,

Toronto, 1977.

[42] Caillet J., Pullman B., In: Molecular

Associations in Biology, Ed. B. Pullman,

Academic Press, 1968.

[43] Handbook of Chemistry and Physics, 52d. Ed.

R.C.Weast, Ohio, 1971.

[44] Golubev A.A., Lublina E.I., Tolokontsev N.A.,

Quantitative Toxicology, Leningrad, 1973.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the Creative

Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.13

Vladimir Mukhomorov

E-ISSN: 2224-2678

133

Volume 21, 2022