Application of the Molecular Pseudopotential Method for Modeling the
Toxicity of Chemical compounds
VLADIMIR MUKHOMOROV
Physical Department
State Polytechnic University
St. Petersburg
RUSSIA
Abstract: - Statistical modeling of the relationship between the toxicity of a number of substituted benzo-2,1,3-
thia- and selenadiazoles depending on the number of various substituents and their position in the benzene ring
was performed. It has been statistically reliably established that the toxicity of the analyzed series of chemical
compounds is closely related to the value of the molecular pseudopotential. It has been shown that the
relationship between the toxicity of drugs correlates linearly with the molecular electronic factor, which
characterizes the magnitude of the pseudopotential of the molecule.
Key-Words: - Toxicity, benzo-2,1,3-thiadiazoles, pseudopotential, statistical modeling, electron factor, trend,
correlation
Received: May 19, 2021. Revised: March 8, 2022. Accepted: April 10, 2022. Published: May 5, 2022.
1 Introduction
The antifungal activity of benzo-2,1,3-thiadiazoles
is known [1], which was revealed on various test
objects. However, there is no single method for
initial trials of antifungal drugs. This makes it
difficult to identify a quantitative relationship
between the chemical structure of compounds and
their toxic effects. Therefore, the method of
modeling the relationship between the bioactivity of
drugs and their molecular structure retains its
relevance in connection with the need to clarify the
mechanism of their action, as well as to predict new
highly active chemical compounds.
2 Problem Formulation
In this article the toxicity of a series of substituted
benzo-2,1,3-thia- and selenediazoles obtained and
tested under the same conditions will be analyzed by
statistical methods [1] as well as a quantitative
relation between the structure of this series of
compounds and their toxic action will be
established. This will allow us to statistically
reliably identify the most significant molecular
parameters of chemical compounds that are
responsible for activation in the biosystem of drugs,
as well as to make some assumptions about the
mechanism of their toxic effects.
3 Problem Solution
The search for the connection between the
molecular structure of a compound and its toxicity is
based on the idea that the objects under study have
some effective electrostatic molecular potential,
which is approximated by pseudopotential. Toxicity
(LD50 in units of mg/kg) [1], the studied series of
compounds is given in Table 1. Thus, the task is to
choose a model with the minimum number of
independent parameters that explain the largest
fraction of the error variance. Determination of the
real molecular potential is associated with complex
quantum chemical calculations, which greatly
complicates the construction of a practically
convenient model. At the same time, the
pseudopotential makes it possible to reliably
reproduce many properties of condensed media. For
example, the model pseudopotential correctly
reproduces the nature of external electron scattering
at atomic potentials in solids. The fact is that the
scattering of an electron on a pseudopotential, which
is rather weak, occurs in the same way as on a true
potential. For a molecule, the pseudopotential is
determined by the sum of the model potentials of
the atoms that form the molecule [2]. It is important
to emphasize that the consequences arising from the
pseudopotential theory are in good agreement with
the known experimental data on electron scattering
[3].
In order to identify the relationship between the
toxic properties of chemical compounds and their
molecular structure, a method is proposed that uses
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the average number of electrons in the outer shell of
atoms in a molecule per atom as a factor sign of a
molecule:
,
(1)
where ni is the number of atoms of the ith type with
the number of electrons in the outer electron shell Zi.
The summation is performed over all atoms in the
molecule; Σni = N is the total number of atoms.
Within the framework of the pseudopotential
method, it was shown [4] that the model
pseudopotential of positive core ions of the
molecule is weakened compared to the Coulomb
field of an isolated core ion due to screening by
external (valence) electrons
(2)
where Z is defined by equation (1); f(r) and F(r) are
corrections [4] to the Coulomb potential, depending
on the distance r between the molecule backbone
and electron; e is the electron charge, RM is the
scattering center radius. It can be shown that the
parameter Z, which characterizes the number of
electrons (2) is a common factor for the
pseudopotential [5]. The model molecular
pseudopotential method assumes that only electrons
in the outer (valence) shell of the scattering center
are taken into account. It is well known that the
chemical properties of molecules are determined by
the electronic state of a relatively small group of
external electrons. The properties of the other
electrons of the atom, which are called frame
electrons, have almost no effect on the physical and
chemical processes in which the molecule
participates. This approximation is sometimes called
the “frozen core approximation”. In this
approximation, the outer electrons do not move in
the real Hartree-Fock force field of the molecule,
but in a much weaker pseudopotential field. In this
case, the behavior of the external electrons is close
to their behavior in the Coulomb electrostatic field,
and the pseudopotential itself is mainly determined
by the first term of the potential (2). The physical
meaning of the model pseudopotential is to describe
the field of the core center in a complex molecular
system or in a solid. The parameter that determines
the potential variations in molecules is the average
number of valence electrons per atom in a molecule.
This result will be used in further studies.
Molecular potential can affect the biological
system by interfering with the mechanisms that
regulate life processes and thereby determine the
biological activity of chemical compounds.
According to the pseudopotential model, the
average number Z of electrons on the outer electron
shells of atoms in a molecule is used as a general
factor (1) and (2) characterizing the molecular
potential.
Fig.1. Molecular structure of substituted benzo-
2,1,3-thia- and selenadiazoles. X = S or Se (see text
for details).
Table 1
Toxicity [1] and molecular factor Z for substituted
benzo-2,1,3-thia- and selenadiazoles.
N
Substitutes
logLD50
Z,
arb.
units
R1
R2
R3
1
NO2
Cl
H
1.80
4.40
2
NO2
Br
H
1.69
5.07
3
NO2
Cl
Cl
1.45
4.80
4
Cl
H
NO2
1.81
4.40
5
Br
NO2
NO2
1.71
5.41
6
H
NO2
H
2.60
4.00
7*)
NO2
Br
H
1.45
5.73
8
OH
NO2
NO2
2.40
4.55
9*)
NO2
H
H
1.71
4.67
10
NO2
H
H
2.61
4.00
11
OC2H5
NO2
NO2
2.25
3.92
12
OC4H9
NO2
NO2
2.30
3.50
13
CH2NH2
H
H
2.78
3.24
14
COOH
H
H
2.78
3.75
15
OH
H
COCH3
2.56
3.47
16
H
OC2H5
H
2.70
3.10
17
NH2
NH2
H
2.78
3.29
18
NH2
CH3
H
2.48
3.11
19
NH2
H
H
2.30
3.33
20
H
NH2
H
2.30
3.33
21
OH
H
H
1.88
3.57
22
H
OH
H
2.36
3.57
23
OH
CH3
H
2.52
3.29
24
Cl
OCH2CO
OC2H5
H
2.78
3.46
25
SH
H
H
2.00
3.57
26
SO2H
H
H
2.70
3.88
27
CH2SP
O(ONa)2
H
H
3.00
3.90
28
H
CH2NH(C
H2)2SO(O)
O(ONa)2H
H
2.78
3.38
29
CH2NH
(CH2)2S2
O3Na
H
H
2.30
3.38
*) Selenium containing compounds.
This article analyzes the cause-and-effect
relationship of toxicity - structure of the molecules
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of a number of diazoles (Fig. 1). From the possible
combination of different causes, an attempt will be
made to identify the most significant, leaving aside
secondary and incidental factors. Let us find out
whether, for example, there is a trend between the
toxicity of sulfur-containing compounds and the
value of the explanatory molecular factor Z.
Let us find out whether, for example, there is a
trend between the toxicity of sulfur-containing
compounds and the value of the explanatory
molecular factor Z. Let us construct a ranked series
according to the toxicity value logLD50 for a sample
composed of mutually independent elements. Each
chemical compound of this series corresponds to a
certain value of the molecular factor Z (Table 1). Let
us determine whether the sequence of Zi values for
the ranked series by toxicity value is random,
unrelated to the value of logLD50 or there is a
systematic component to this sequence. To do this,
we will use the Abbe-Linnik test [6,7]:
Q* = - (1 – q)∙[(2N + 1)/(2 – (1 – q)2)]0.5
= - 4.33 < u0.05 = - 1.645. (3)
Since q < qcr and Q* < u0.05, the null-hypothesis of
series randomness Zi is rejected and the alternative
hypothesis is accepted, which indicates a systematic
shift of the mean [7] with confidence probability
0.95. It follows from inequalities (3) that the
sequence of Zi values has the following connection:
the greater the value of the explanatory molecular
factor Zi, the more toxic the chemical compound. An
approximate (for a sample size of less than 50) value
of the parameter Q* also indicates the existence of a
systematic shift of the mean value. Using the
toxicity data as well as the numerical values of the
factor Z from Table 1, the corresponding scatter
diagram can be constructed (Fig. 2). Scattering can
be caused by unaccounted for factors (not
necessarily secondary) or by chance, while the
relationship becomes stochastic. Figure 2
demonstrates homoscedasticity - the relative
stability and homogeneity of the random error
variance of the regression model. Statistics (3) and
the location of points (initial data) on the scatterplot
suggests that there is a clear trend between the
explanatory molecular factor Z and the resultant
sign.
Fig.2. Scatterplot of toxicity observations (logLD50)
for benzo-2,1,3-thia- and selenediazole derivatives.
The regression line is determined by equation (4). ●
- sulfur-containing preparations, ▲ - selenium-
containing preparations.
Indeed, using the methods of statistical analysis, it
was found that between the explanatory sign Z and
the toxicity (logLD50) of chemical compounds, there
is the following averaged and statistically significant
negative linear relationship:
logLD50mod(Z) = b0(1) + b1(1)Z, N = 29, standard error
of the regression estimate: S1 = 0.3125; R1 = -0.73 ±
0.09; with a significant relationship between the
explanatory variable and the resulting variable, the
correlation coefficient should be significantly
different from zero: |R1| > R0.05cr(f – m – 1) = 0.367
[8]; sample size sufficient for the significance of the
correlation coefficient: N0.05min = 7 [9]; b0(1) = 4.10
± 0.33, b1(1) = - 0.46 ± 0.08, t(b0(1)) = 13.34 > |
t(b1(1))| = 5.49 > t0.05cr(N – 2) = 2.052, F = 30.15 >
F0.05cr(f1 = 1; f2 = 27) = 4.21; Δ = N-1∙Σi(|lgLD50i -
lgLD50imod|∙100%/ lgLD50i) = 11%.
(4)
The significance of the regression coefficients bi (4)
is tested using Student's t-distribution (two-sided
critical area) for N – 2 degrees of freedom and at a
significance level of α. If |t(b)| > tcr, then the
regression coefficient b is significantly different
from zero at the 95% confidence level. Since F >
F0.05cr(f1; f2), we can recognize the significance in
general of the regression equation at α = 0.05. The
value of F is related to the coefficient of
determination as follows [10]: F = R2(N – 2)/(1 –
R2). According to the Cheddock scale [12], if the
3 4 5 6
1
1.5
2
2.5
3
Z arb. units
logLD50
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correlation coefficient falls into the range of values
0.7 < |R*| < 0.9, then the relationship between the
variables is characterized as "strong (close)
connection". For small samples, it is recommended
[8] to use the corrected correlation coefficient: R* =
R∙[1 + 0.5∙(1 – R2)/(N – 3)]. Using the data [9], you
can specify the minimum sample size N0.05min = 7 <
N = 29, sufficient for the reliability of the
correlation coefficient |R*| = 0.74 at the 95%
confidence level.
Confidence limits of unit toxicity prediction
log(LD50) for simple linear regression:
(5)
Here tα(f) is the quantile of the t-distribution with f
= N - m - 1 degrees of freedom and significance
level α; m is the number of explanatory variables.
The value S(Z) for the tested value Z can be
calculated by the formula:
(6)
here Sser is the standard error of the residuals
Sser = (7)
To determine a statistically significant linear
correlation coefficient, it is necessary to fulfill the
requirement of homogeneity of the analyzed sample
[10]. It can be shown that the set of elements of
logLD50 and Z (Table 1) are homogeneous and have
a distribution close to the normal distribution at the
95% confidence level.
Population statistics of logLD50:
N1 = 29, logLD50av = 2.34 ± 0.09; (2.16 - 2.51) is the
confidence interval at significance level α = 0.05;
logLD50min = 1.45, logLD50max = 3.00; standard
deviation: SlogLD1 = 0.46; τmax = 1.44 < τmin = 1.93 <
τ0.05cr(N1) = 2.94; Wilk-Shapiro normality test: W =
0.918 ≈ W0.05cr(N1) = 0.928, David-Hartley-Pearson
normality test: U10.05cr(N1) = 3.47 ≈ U =
[(logLD50max – logLD50min)/SlogLD] = 3.37 <
U20.05cr(N1) = 4.89 ; the coefficient of variation: V
= (19.66 ± 2.58)%; representativeness of the sample
size [9]: N1repr = 23;
population statistics of the factor Z:
N1 = 29, Zav = 3.90 ± 0.13; (3.63 - 4.17) is the
confidence interval at significance level α = 0.05;
Zmin = 3.10, Zmax = 5.73, SZ1 = 0.704, τmin = 1.14 <
τmax =2.60 < τ0.05cr(N1) = 2.94; Wilk-Shapiro
normality test: W = 0.898 ≈ W0.05cr(N1) = 0.928,
David-Hartley-Pearson normality test: U10.05cr(N1) =
3.47 < U = [(Zmax – Zmin)/SZ] = 3.73 < U20.05cr(N1)
= 4.89 ; V = (18.05 ± 2.37)%; N1repr = 23.
(8)
The homogeneity of the sample depends only on the
sample size and is determined by the critical value
of the Grubbs-Romansky τ-test [6,11]. If the
volumes of samples Z and logLD50 are sufficient for
the reliability of determining their main statistical
indicators - mean values and standard deviations,
then the volumes of these samples are also usually
sufficient to identify the trend of the relationship
between them [9].
The regression equation for the sample N2 = 27,
which includes only sulfur-containing drugs,
practically does not differ from the regression (4):
logLD50mod(Z) = b0(2) + b1(2)Z, N2 = 27, R = -0.65 ±
0.12, |R*| = 0.68 > R0.05cr(N2 – 2) = 0.381; sample
size sufficient for the significance of the correlation
coefficient: N0.05min = 9; b0(2) = 4.02 ± 0.36, b1(2) =
- 0.44 ± 0.10, t(b0(2) ) = 10.12 > |t(b1(2))| = 4.24 >
t0.05cr(N2 – 2) = 2.060, RMSE(S2) = 0.321; F = 18.0 >
F0.05cr(f1 = 1; f2 = 25) = 4.24. (9)
The value of the empirical correlation ratio ηemp =
0.679 was determined for the relationship
logLD50mod(Z) at N2 = 27. Initial data after ranking
by Z were divided into five groups: n1 = 5, n2 = 5, n3
= 6, n4 = 6, n5 = 5. Intergroup variance Slg2 = N-1∙
Σi(logiav – logav)2∙ni = 0.078 and total variance Slg2 =
N-1∙Σj(logj)2 – (logav)2 = 0.169, respectively. Here
logiav is the values of the group means (options of
the feature logLD50); logav is the overall average
value of the response function logLD50; index i =
1,2,…,5; index j = 1,2,…,27. In accordance with the
Blackman curvilinearity criterion [9], we obtain the
following inequality: Bl = N2∙|ηemp2 – R2| = 1.05 <
Blcr = 11.37, which indicates that the relationship
between features should be straightline. An estimate
of the curvilinearity of the relationship can also be
obtained from the following relation:
t = 0.5∙N0.5[(ηemp2 – R2)-1 – 2 + ηemp2 + R2]-0.5 =
0.523 < 3.00. (10)
Since the inequality (10) is satisfied, we can agree
that the analyzed relationship may slightly deviate
from the straight-line dependence. The reliability of
the correlation relationship is checked by the
following relationship:
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F =
(N2 – n)/(1 –
)/(n – 1) = 4.71 >
F0.05cr(f1 = n – 2; f2 = N2 – n) = 3.05. (11)
A criterion based on Fisher's normalizing z-
transformation [8]: u = 0.825 > u0.05(N2) = z0.975∙(N2
– 3)-0.5 = 0.742 also indicates the significance of the
correlation coefficient. To check the difference
between the coefficients b1(1) (4) and b1(2) (9), we use
the following relationship [10]:
t = |b1(1) - b1(1)|∙
= 0.216 < t0.05cr(N1 + N2 – 4) = 2.006, (12)
which is valid because the following inequality
holds (the ratio of the larger variance to the smaller
variance): F = (S2/S1)2 = 1.051 < F0.05сr(f1 = 25; f2 =
27) = 1.93. The inequality (12) quantitatively
indicates the absence of a statistically significant
difference between the regression coefficients b1(1)
and b1(2). That is, the addition of selenium-
containing compounds to a particular sample does
not change the slope of the straightline regression
(4). Additional information about the presence of a
systematic shift in the average explanatory factor Z
for the ranked toxicity of chemical compounds (e.g.,
only for sulfur-containing drugs; sample size N =
27) can be obtained by using the Abbe-Linnik test
(3):
q = 0.2578 < q0.05cr(N = 27) = 0.6996,
Q* = - (1 – q)∙[(2N + 1)/(2 – (1 – q)2)]0.5 =
- 4.73 < u0.05 = - 1.645. (13)
The smaller the empirical value q in comparison
with the critical value, at the chosen level of
significance, the clearer the relationship between the
dependent feature and the explanatory variable. To
make a statistical conclusion about the existence of
a correlation, it is necessary to check the
significance of the sample pair correlation
coefficient. If there is a connection between the
explanatory and resulting variables, the correlation
coefficient R should be statistically significantly
different from zero. The null hypothesis of no
relation between the variables can be rejected if
Fisher's inequality for t-distribution with f = N - m -
1 degrees of freedom is fulfilled for a sample
correlation coefficient R = - 0.74: tF = |R|(N – m –
1)0.5/(1 – R2)0.5 = 5.72 > t0.05cr(N – 2) = 2.052. Here
m = 1 is the number of explanatory variables. If the
t-statistics calculated from the results of the sample
is such that tF < t0.05cr(f), then a null hypothesis is
accepted at a significance level of α = 0.05, and the
deviation of the correlation coefficient R from zero
can only be attributed to unaccounted or random
variations. For the studied relationship, the
inequality tF > t0.05cr(f) was obtained, therefore, there
is a significant statistical relationship between the
variables. Note that a two-sided critical region is
used here.
Let us use regression (4) to estimate the expected
toxicity of a chemical compound that was not
included in the original sample. Known [13]
observed toxic dose logLD50 = 2.48 for
unsubstituted benzothiadiazole (gross formula
C6H4N2S; Z = 3.39 arb. units). Substituting the value
Z = 3.39 arb. units into regression equation (4), we
obtain the following toxicity estimate logLD50mod =
2.58 for benzo-2,1,3-thiadiazole, which is close
(comparison error < 4%) to the observed value of
2.48.
4 Conclusion
Checks were made on the use of additional
explanatory variables in the regression equation.
The Gammet constant σm of substituents in position
R1 of the benzene ring, the molar refraction MR(R1),
which characterizes the volume size of substituents,
as well as the contribution of π, which determines
the hydrophobicity of substituents, were taken into
account [14]. Taking these indicators into account
did not improve the quality of the regression, and
their contribution to the regression equation was
statistically insignificant. However, it can be noted
that those chemical compounds for which the
substituents in the R1, R2, and R3 positions
preferably have a high electron affinity (i.e., they are
electron acceptors) at the same time have the
greatest toxicity. For example, replacing the
hydrogen atom in position R3 for the molecule (no.
1 in Table 1) with a chlorine atom (no. 3)
significantly increases the toxic properties of the
drug. A similar situation occurs when comparing
molecule no. 5 with molecules nos. 8, 11, and 12.
The electron affinity of the bromine atom in position
R1 is almost two times higher than that of the OH
and OC2H5 substituents and 4–5 times higher than
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that of the OC4H9 substituent. According to the
energy of electron affinity, the substituents can be
arranged in the following sequence: NO2 > Cl > Br
≥ SH > OH > OC2H5 > NH2 > OC4H9 > H [16].
As the explanatory factor Z increases, there is a
tendency for the toxicity of chemical compounds to
increase, and this tendency has a statistically
significant linear character. Deviations from the
regression line can be attributed to the influence of
other unaccounted factors or random fluctuations.
Apparently, the molecular potential of benzo-
2,1,3-thia- and selendiazole derivatives
approximated by pseudopotential (1) - (2)
determines the possible ability of chemical
compounds to enter into paired intermolecular
interaction with some region of the biophase and
thereby initiate the toxic action of the drug. The
greater the value of the molecular factor Z, the
stronger the pairwise interaction of the molecule
with the biophase region [16].
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