Simulation of the Radioprotective Action of Mercaptoethylamine
Derivatives and its Analogues with their Quantum-Chemical and
Information Features
MUKHOMOROV V.K.
Physical Department
State Polytechnic University
St. Petersburg
RUSSIA
Abstract: - Quantum chemistry, condensed matter physics and applied information theory methods are used to
reveal the relationship between the molecular structure of radiation injury modifiers and their radioprotective
activity in the series of aminothiols and their analogues. Significant electronic and informational parameters of
molecules, which are associated with the radioprotective effect of drugs, were determined by statistical analysis
methods. Based on the identified significant molecular parameters, possible mechanisms of the biochemical
and biophysical radioprotective action of the analyzed chemical compounds are discussed. The detected
significant molecular parameters suggest what possible molecular processes these drugs can take part in and
what electronic and informational properties radioprotectors molecules should possess.
Key-Words: - Aminothiols, radioprotectors, electronic energy, threshold, pseudopotential, dipole moment,
information function, modeling
Received: May 30, 2022. Revised: October 11, 2022. Accepted: November 14, 2022. Published: December 31, 2022.
1 Introduction
The problem of changing the body's radio-
sensitivity through the use of various chemical
compounds continues to be one of the most topical
and intensively developed in modern radiobiology.
The study of molecular mechanisms of action of
radiation lesion modifiers is of fundamental
importance for understanding the triggering effects
of radiation and mechanisms of radiation protection.
At the same time, deciphering the molecular
mechanisms of radiation exposure opens up the
prospect of new approaches to the search for
effective radio-protective agents. In this connection,
of considerable interest are studies concerning the
connection between the radioprotective effect of
drugs and the electronic structure of molecules and
their informational content. Aminothiols are of great
interest due to their diverse applications in medicine
and organic chemistry. Aminothiols are active
fungicides and have an antibacterial effect, exhibit
herbicidal activity and antidote properties, as well as
antihemolytic and hypotensive effects [1]. In this
article, quantitative relationships will be obtained
between the features that determine the energy and
information properties of the molecules of
mercaptoethylamine derivatives and its analogues
and their radio-protective effect. The antiradiation
properties of these preparations have been studied
experimentally in sufficient detail.
2 Problem Formulation
Obviously, modification of the basic molecular
structure is one of the ways to influence the
molecular factors that determine the radioprotective
efficacy of the drug. It is of some interest to reveal
the cause-and-effect relationship between the radio-
protective activity of molecules of a number of
mercaptoethylamine derivatives and their analogues
(Table 1) under conditions of varying the molecular
structure, which are accompanied by changes in
electronic and information properties of molecules.
To activate the protective action of the drug,
apparently, it is necessary for the molecule to
interact with the active centers of the biosystem.
This can lead to a restructuring of some
physiological processes of the body accompanied by
an increase in its radioresistance. The molecule of
radioprotector can interact with biologically
important macromolecules that are sensitive to the
action of radiation. Radiation damage to the
biosystem is complex and consists of both the act of
excitation or ionization and a number of
accompanying rapid processes, such as the
migration of charge and energy of secondary
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electromagnetic radiation, localization of charge,
polarization of the medium, etc. [2]. One of the
conditions for the repair of such damage is the
contact of the damaged molecule with an exogenous
impurity molecule, as well as the presence of charge
and energy migration paths between them. That is,
there must be long- and short-range (on the
molecular scale) interactions leading to the
formation of relatively stable intermolecular
complexes including biomacromolecules and low-
molecular impurities. The excess energy received by
biomolecules as a result of irradiation can be used to
destroy the intermolecular bonds stabilizing the
complexes. It is possible that the weakening of the
effect of irradiation on the body is associated with
the creation of obstacles to the implementation of
radiation damage by hindering the electronic-
conformational transformations of macromolecules.
It is known [3] that after irradiation in the presence
of effective aminothiol radioprotectors, modification
of single- and double-stranded DNA breaks occurs.
3 Problem Solution
In this article, various possible manifestations of
primary physical and chemical processes at the
molecular level, arising under the influence of high-
energy radiation, will be compared with the
quantum chemical characteristics of molecules.
Knowledge of the electronic structure of molecules
makes it possible to apply this information in the
analysis of various ideas about the mechanisms of
the protective action of low-molecular compounds.
That is, it is possible to identify which quantum
parameters of molecules are the most common and
informative for the analyzed series of chemical
compounds.
3.1 Electronic properties of molecules
The electronic characteristics of the substituted
aminothiols and their analogues were calculated
using the semiempirical Hartree-Fock self-
consistent field method in the MINDO/3
approximation [4], taking into account the
optimization of the spatial geometry of the
molecules. The method provides satisfactory results
for most standard characteristics of molecules,
including molecules containing phosphorus and
sulfur atoms. Analysis of the electronic features of
the molecules of this series of chemical compounds
showed that the most informative molecular
parameters are the boundary one-electron molecular
orbital (MO) energies: the highest occupied εoc, the
lowest unoccupied εun spin orbitals, the energy
interval Δε = εun - εoc, as well as the squares of
dipole moments of molecules μ2. Table 1 shows the
calculated values of εoc, εun, μ and Δε, as well as the
radio-protective trait - survival rate (A, %), of
irradiated mice at absolutely lethal dose [5,6]. It is
well known that the energy of the highest occupied
molecular orbital of an isolated molecule determines
its ionization potential (in accordance with
Koopmans theorem for molecules with closed
shells). The energy interval Δε approximates (the
electronic transition is limited by the symmetry of
the one-electron levels) the electronic excitation
energy of an isolated molecule. Since the results of
biological effect assessment of drugs depend, in
general, on many different factors that cannot
always be taken into account, it is convenient for
further statistical analysis to divide all chemical
compounds presented in Table 1 into three groups
according to the result indicator (А): highly active
(A1 survival rate ≥ 60%; relatively low doses),
medium active (A2 = 50%; medium doses) and
slightly inactive or inactive drugs (A3 ≤ 30%; high
doses). Table 1 shows either the protection range or
the maximum possible protection. The protection
effect depends significantly on the applied dose of
the drug [5] and is limited by various factors,
including the toxicity of the drugs. Table 1 shows
either the protection range or the maximum possible
protection. Using the results of quantum-mechanical
calculations (Table 1), we divide all chemical
compounds on the basis of Δε into three groups. The
first group includes preparations for which the value
of Δε < 8.5 eV. The second group contains chemical
compounds for which the energy difference is in the
relatively narrow range of 8.5 eV ≤ Δε ≤ 9 eV. The
third group includes preparations for which the
difference Δε > 9 eV. The numerical material can
now be presented in the form of a 3×3 contingency
table (Table 2). In this case, features A and Δε can
be called interval features. Following the method
detailed in [7,8], the empirical values qij in Table 2
determine the frequencies of occurrence of sign
values in admissible areas (1, 2 and 3) defined by
interval signs A and Δε. If there were a one-to-one
relationship between the attributes, non-zero values
would only be on the diagonal of the table.
Verification of the statistical reliability of the
relationship between the radioprotective effect of
substituted aminothiols and the energy value Δε is
performed by comparing the empirical values of
frequencies qij with the expected values. We choose
the relative frequencies qij’ as the expected values,
so that the distribution over the cells of the table
would correspond to the absence of connection
between the events.
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Table 1
Electronic and information features of substituted aminothiols and their analogues, as well as their bioactivity
No
Chemical compounds
εoc
∆ε
μ,
D
Z,
conv.
units
H,
bits
dH1,
bits
Aexp, %
eV
1
H2N(NH=)CNHCH2CH2SPO3H2
-9.43
4.73
4.48
3.140
2.130
-0.109
100
2
H2NCH2CH2SC(=NH)NH2
-7.63
7.82
2.53
2.625
1.623
-0.014
100
3
H2NC(=NH)CH2SH
-8.25
7.06
2.48
2.727
1.686
-0.030
100
4
H2N(NH=)CNHCH2CH2SH
-7.53
7.26
4.38
2.625
1.623
-0.014
100
5
H2NCH2CH2SPO3H2
-9.57
4.89
4.89
3.125
2.078
-0.125
100
6
H2NCH2CH2SSO3H
-9.25
6.31
4.82
3.333
2.013
-0.126
100
7
(CH3)2N(NH=)CCH2SH
-7.87
7.85
2.42
2.471
1.545
0.041
100
8
H2NC(=NH)CH2CH2SH
-7.87
7.84
3.11
2.572
1.611
0.015
100
9
H2NC(=NH)CH2SSO3H
-8.52
5.52
3.65
3.600
2.156
-0.141
100
10
H2NC(=NH)CH2SPO3H
-8.66
4.08
4.86
3.375
2.225
-0.147
100
11
H2NCH2CH2SC(=S)SH
-8.63
7.04
2,56
3.000
1.725
-0.024
100
12
H2NC(=NH)CH2SSCH2C(=NH)NH2 (T)*)
-7.78
6.56
0.94
2.900
1.761
-0.036
100
13
CH3C(NH2)HCH2SH
-7.38
7.18
0.91
2.286
1.430
0.066
70
14
H2NCH2CH2SH
-8.32
8.16
2.71
2.364
1.491
0.032
60
15
(CH3)2S=O
-9.39
7.21
3.59
2.600
1.571
0.022
65
16
H2NCH2CH2SCN
-8.89
8.17
1.98
2.833
1.729
0.000
50
17
H2NCHCOOHCH2SH
-9.27
7.84
2.57
3.000
1.921
-0.024
50
18
H2NCH2CH2SC6H5
-8.38
8.30
2.13
2.571
1.438
0.042
50
19
H2NCH2CH2CH2SH
-8.85
9.35
2.41
2.286
1.430
0.066
50
20
H2NC(=NH)SCH3
-8.81
8.03
2.85
2.462
1.547
-0.016
50
21
CH3CH(SH)CH2NH2
-8.91
9.03
2.56
2.286
1.430
0.066
50
22
H2NCH2CH2SC(=O)CH3
-8.88
8.84
1,30
2.733
1.774
0.025
50
23
H2C=CHCH2NHCH2CH2SH
-8.61
8.75
2.43
2.333
1.411
0.079
50
24
CH3CH2SC(=NH)NH2 (T)
-8.64
8.97
2.49
2.572
1.611
0.015
50
25
H2NCH2CH2SСH3
-8.89
8.83
2.31
2.286
1.430
0.066
30
26
H2NCH2CH2SСH2CH3
-8.90
8.73
2.67
2.235
1.379
0.085
20
27
H2NCH2CH(SH)COOH (T)
-9.29
8.54
2.56
3.000
1.921
-0.024
10
28
H2NCH2CH2SCH2CH=CH2
-8.89
8.82
2.30
2.333
1.411
0.079
0
29
H2NCH2CH2CH2CH2SH
-8.85
9.30
2.59
2.235
1.379
0.085
10
30
OHCH2CH2SH
-9.80
9.72
2.05
2.600
1.571
0.022
0
31
(CH3)2NCH2CH2SH
-8.28
8.27
1.09
2.235
1.379
0.085
10
32
H2NCH2C(CH3)2SH
-8.88
8.94
2.52
2.375
1.424
0.076
10
33
H2NC(=O)NHCH2CH2SH**)
-9.40
9.32
2.24
2.800
1.857
-0.019
25
34
H2NCH2CH2OH
-8.98
10.7
1.87
2.364
1.491
0.032
0
35
CH3CH2CH2SH
-9.44
9.85
2.49
2.167
1.189
0.110
10
36
CH3NHCH2CH2SH (T)
-8.55
8.67
2.50
2.286
1.430
0.066
10
37
H2NCOCH2SH
-9.85
9.43
2.49
3.000
1.961
-0.036
10
38
H2NCH2COSH
-9.37
9.45
1.66
3.000
1.961
-0.036
0
39
H2NCOCH2CH2SH
-9.55
9.59
2.08
2.769
1.823
0.007
0
40
H2NOCH2CH2SH
-8.39
8.45
2.69
2.667
1.781
-0.023
10
41
OHCH2CH2SC(=NH)NH2
-8.74
8.91
2.49
2.800
1.857
-0.019
0
42
H2NCH2CH2COSH
-9.02
9.48
1.79
2.769
1.823
0.007
0
43
H2NCOCH2CH2SC(=NH) NH2
-8.89
8.88
1.76
2.889
1.877
-0.018
0
44
H2NCH2CH(OH)CH3
-8.95
10.7
1.81
2.625
1.623
-0.014
0
45
H2NCH2CH(Cl)CH3
-8.88
9.86
1.15
2.462
1.489
0.057
0
*) The T index indicates that the drug is toxic. **) With oral administration of the drug at a dose of 1000 mg/kg, there is no
protection.
Obviously, in the case of independent events, the
joint proportion (relative frequency) is equal to the
simple product of the proportions: pij = pi∙Pj ≡
qi∙Qj/N2; here pi = qi/N. The numerical values of qi
and Qj are given in Table 2. The theoretically
expected relative frequencies qij' are determined as
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follows: qij’ = N∙pij ≡ qi ∙Qj/N. (1)
Such values of frequencies would occur in the
absence of a connection between the events. The
chi-square test is used to test the null-hypothesis
that there is no relation between the events.
Table 2
Relationship between the radioprotective effect of
substituted aminothiols and their analogues and the
value of the electronic sign Δε.
A,
%
Characteristic Δε (in eV)
Δε < 8.5
8.5≤ Δε
≤ 9.0
Δε > 9.0
Total
≥ 60
q11 = 15
q11' = 7
q12 = 0
q12' = 3.67
q13 = 0
q13’ = 4.33
q1 = 15
p1 = 0.333
q1’ = 15
= 50
q21 = 4
q2' = 4.2
q22 = 3
q22’= 2.2
q23 = 2
q23’ =2.6
q2 = 9
p2 = 0.200
q2’ = 9
≤ 30
q31 = 2
q31' = 9.8
q23 = 8
q23' = 5.13
q33 = 11
q33’ = 6.07
q3 = 21
p3 = 0.467
q3’ = 21
Q1 = 21
P1 = 0.467
Q2 = 11
P2 = 0.244
Q3 = 13
P2 = 0.289
N =45
3
1ii
P
=
3
1jj
p
=1.
If the chi-square value is less than the table value
at a given level of significance and number of
degrees of freedom, then the null-hypothesis (no
relation between the signs) is accepted. Using the
data in Table 2, we obtain the following inequality:
χ2 = ∑(qij – qij’)2/qij’ = 29.4 >χ0.05cr,2 (f = 4) = 9.488.
(2)
Here the summation is performed over indices i and
j from 1 to 3. Number of degrees of freedom f = (v –
1)∙(w – 1) = 4; here v is the number of rows, w is the
number of columns. For a 3×3 contingent table
(Table 2) v = w = d = 3. Thus, inequality (2) with a
probability of 0.95 allows us to reject the null-
hypothesis and accept that the events A and Δε are
significantly interconnected. The value of the
energy interval is associated with the radio-
protective effect of drugs. This conclusion is also
preserved when choosing the significance level α =
0.001, i.e. 0.1%. Consequently, a decrease in the
energy difference Δε is accompanied by an increase
in the radioprotective effectiveness of chemical
compounds of a number of aminothiols and their
analogues. Obviously, if there were an absolute
unambiguous relationship between features, then the
table should contain only diagonal elements.
The measure of the strength of the relationship
between events can also be quantified using
Pearson's contingency coefficient (0 ≤ K ≤ 1) and
Chuprov's contingency coefficient ϕ (0 ≤ ϕ ≤ 1))
[8]: K = {χ2∙d/[( χ2 + N)(d – 1)]}0.5 = 0.77,
ϕ ={χ2/[N∙(d – 1)]}0.5 = 0.57. (3)
Here d = 3, that is, the number of rows and columns
of a 3×3 table is the same. Both the coefficients K
and ϕ indicate the existence of a strong connection
between the events. It is usually assumed that the
relationship between signs is close if the inequalities
K ≥ 0.5 and ϕ ≥ 0.3 are satisfied. Pearson's
contingency coefficient is comparable to the linear
correlation coefficient, which in this case is |r| =
0.82. Let's check whether the average values of Δεav
differ significantly for the three areas of activity: A1
≥ 60%, A2 = 50% and A3 ≤ 30%. For regions A1 and
A2 we obtain the following sample statistics of
average values:
N1 = 15, Δε1av = 6.63 ± 0.33; 95% confidence
interval: (5.93-7.34), Δε1min = 4.08, Δε1max = 8.16, S1
= 1.27; Grubbs-Romanovsky homogeneity test for
small samples: τmax = 1.20 < τmin = 2.01 < <
τ0.05cr,2(N1) = 2.493 < τ0.05cr,1(N1) = 2.617; the Wilk-
Shapiro normality test: W = 0.894 > W0.05cr(N1) =
0.881; the David-Hartley-Pearson normality test [9]:
U10.05cr(N1) = 2.97 < U = [(Δε1max – Δε1min)/S1] =
3.21 < U20.05cr(N1) = 4.17; coefficient of variation:
V1 = S1∙100% = 19.2%; δV1= ± V1/(2N1)0.5 = ±
3.56%; accuracy of experience: P1 = V1/N10.5 =
4.96%; N1repr = 12; according to [10] the
representativeness of the sample arithmetic mean is
also determined by the inequality: Θ = y1∙[(N1 –
1)/(N1∙y2 – y12)]0.5 = 20.23 > Θcr = 3,
here we use the following notations: y1 is the sum
of the variants of the series, y2 is the sum of the
squares of the variants of the series;
N2 = 9, Δε2av= 8.58 ± 0.17; 95% confidence interval:
(8.19-8.98), Δε2min = 7.84, Δε2max = 9.35, S2 = 0.517,
τmin = 1.44 < τmax = 1.478 < τ0.05cr,2(N2) = 2.237<
τ0.05cr,1(N2) = 2.392; the Wilk-Shapiro normality test:
W = 0.946 > W0.05cr(N2) = 0.829, the David-Hartley-
Pearson normality test: U10.05cr(N2) = 2.59 < U =
[(Δε2max – Δε2min)/S2] = 2.92 < U20.05cr(N2) = 3.552;
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V2 = (6.02 ± 1.42)%. P2 = 2.01%; N2repr = 8: Θ =
49.8. (4)
It follows from inequalities (4) that the
populations Δε1 and Δε2 are homogeneous, and the
elements of the populations are normally
distributed. We first need to check whether the
variance of the residuals in the two populations is
different. To do this, let's calculate the ratio of the
larger variance to the smaller variance. This relation
has an F distribution, which should be compared to
the table value:
F1,2 = S12/S22 = 6.03 >
F0.05cr(f1 = N1 – 1:f2 = N2 – 1) = 3.23. (5)
Since F > F0.05cr, the variances are significantly
different and the comparison of the mean values
should be performed using the following
relationship:
t = Δε2av – Δε1av = 1.95 > Tav =
[v1∙t0.05cr(f1)+v2∙t0.05cr(f2)]/(v1 + v2)0.5 = 0.66, (6)
where v1 = S12/N1 and v2 = S22/N2; number of
freedom degrees: f1 = N1 – 1, f2 = N2 – 1. Here the
one-sided Student's test is applied. Since inequality
(6) holds, the hypothesis of the equality of the mean
values can be rejected. It follows from inequality (6)
that at the 95% confidence level, the average values
of the energy interval Δεav for the regions A1 and A2
are significantly different and this difference is not
random. Using the results (4) we can also calculate
the biserial correlation coefficient between the first
and second groups [11]:
rbs = [(∆ε2av – ∆ε1av)/S]∙[N1∙N2/(N2 – N)]0.5 = 0.663,
t = 4.15 > t0.05cr(N – 2) = 1.72,
S2 = [(N1 – 1)∙S12 + (N2 – 1)∙S22]/(N – 2), (7)
which is significant at the 95% level; here the total
N = N1 + N2 = 24; standard deviation S = 1.417.
Now let's compare the areas of bioactivity A2 and
A3. The population statistics Δε3 for area A3 is as
follows:
N3 = 21, Δε3av = 9.26 ± 0.15; 95% confidence
interval: (8.96-9.56), Δε3min = 8.27, Δε3max = 10.7, S3
= 0.664, τmin = 1.49 < τmax = 2.17 < < τ0.05cr,2(N3) =
2.644 < τ0.05cr,1(N3) = 2.750; the Wilk-Shapiro
normality test: W = 0.933 > W0.05cr(N3) = 0.918, the
David-Hartley-Pearson normality test: U10.05cr(N3) =
3.18 < U = [Δε3max – Δε3min)/S3 ] = 3.66 < U20.05cr(N3)
= 4.49; V2 = (7.17 ± 1.11)%, P2 = 1.56%, N3repr =
17; Θ = 7.2. (8)
Since the difference between the variances for
regions 2 and 3 of bioactivities is not significant:
F = S32/S22 = 1.65 >
F0.05cr(f3 = N3 – 1:f2 = N2 – 1) = 3.15, (9)
then the comparison of average Δεav values for
regions A2 and A3 should be performed using the
relation [11,12]:
t = Δε3av – Δε2av = 0.67 > tav =
t0.05cr(f = N23 – 2)∙{N23S232/[N2N3(N23 – 2)]0.5 = 0.42,
N23 = N2 + N3, S232 = (N2 – 1)∙S22 + (N3 – 1)∙S32.
(10)
Inequality (10) also holds when using the two-
sided criterion t0.975cr = 2.05. Thus, there is a
significant difference between the average values
for the energy interval Δε for the two neighboring
regions of bioactivities A2 and A3, and the following
inequalities are observed: Δε3av > Δε2av > Δε1av.
Consequently, there is a trend in the relationship
between the bioactivity of chemical compounds and
the value of the energy interval Δε. The smaller the
value of Δε, the lower the energy required to excite
an isolated molecule is likely to be. That is, it can be
assumed that, in accordance with this sequence, the
electron-donor properties of the molecule are
enhanced. At the same time, since Δε is defined as
the difference between the MO energies, the
decrease in the difference Δε can be associated with
a decrease in the energy scale of the molecular level
εun. This, in turn, leads to an improvement in the
acceptor properties of the molecules. According to
Szent-Györgyi [13], molecules with a low Δε value
are catalytic electron transmitters and have both
good donor and acceptor properties. As is known,
the decisive factor in the protective effect of sulfur-
containing preparations is the accumulation of
radioprotector molecules up to their threshold
concentration in the cells of critical organs of the
body. The donor-acceptor interaction (a mechanism
caused by the exchange of electrons between the
filled orbitals of the donor molecule and the vacant
orbitals of the acceptor molecule) leads to the
binding of the drug in the body. The resulting
energy level of the complex lies below the initial
states. The resulting binding energy level of the
complex lies below the initial states. Delocalization
of electron leads to the formation of a molecular
complex. For the donor-acceptor mechanism of
complex formation, the position on the energy scale
of the vacant molecular orbital is important. In the
works [3,14] it has been proved that aminothiols as
radioprotectors have the ability to form temporary
mixed disulfide bonds with the enzymes responsible
for the synthesis of DNA precursors. The ability of
aminothiols to interact with proteins can lead to
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short-term blocking of metabolic processes,
including DNA synthesis. The transfer of an
electron between molecules is characteristic of
many fundamental biological processes, which are
accompanied by the formation of complexes with
charge transfer.
Let us also perform an additional check for the
presence of a systematic shift in the average
molecular factor Δε. To do this, we will use the
Abbe-Linnick test [15,16] for the sequences of
bioactivities in Table 1 ordered by magnitude. For a
sample of independent, normally distributed random
variables Δε the trend hypothesis is tested by the
following statistics:
N
ii
N
iii
q
1
2
av
1
1
2
1/5.0
= 0.306 < q0.05cr(N) = 0.7605,
Q* = - (1 – q)∙[(2N + 1)/(2 – (1 – q)2)]0.5 = - 5.37.
Here Δεav = 8.25 is the arithmetic mean value;
sample volume N = 45. The approximate statistical
index Q* is preferably used for sample sizes N ≥ 60.
However, the resulting estimate for smaller sample
sizes usually does not contradict the inequality for
the statistical test q; Q* has a standard normal
distribution [15]. If q > qcr, then we can assume that
the observations do not contain a systematic shift of
mathematical expectations. Since the reverse
inequality q < qcr is satisfied, and the inequality Q* =
-5.37 < u0.05 = -1.645 is also valid, then the null-
hypothesis about the equality of the means of the Δεi
series is rejected (an alternative hypothesis about the
presence of a systematic bias is accepted) with a
probability of 0.95, in this case; up/2 is the quantile
of the normal distribution at p = 0.10. Thus, an
increase in the energy interval Δε is associated with
a decrease in the value of the effective feature Aexp.
Statistical methods are usually used in a complex
manner, due to the complexity of the processes
under study. One of the most common methods of
applied statistics is regression analysis, which is
used to determine the functional relationship
between the resulting factor and many possible
explanatory variables. In this case, the explanatory
variables are related to the bioresponse by some
regression function. However, as is known [11,17],
correlation analysis establishes only the strength of
the connection. The analysis showed that the
relationship between the radioprotective activity (A,
%) of aminothiols from Table 1 and the value of
electronic energy Δε can be approximated by the
following empirical non-linear dependence (Fig. 1):
A(Δε) = 1/[1+c∙exp(b0 + b1∙Δε)], N = 45. (11)
Hereinafter it is assumed that A ≡ A(in
percent)/100%. Using the Grubbs-Romanovsky τ-
test, it can be shown that the initial data for the
radioprotective efficacy of drugs satisfy the
uniformity condition:
.12.3)(12.1
,12.3)(40.1
/||
cr
05.0
min
cr
05.0
max
avminmax/
N
N
SAA
(12)
Here Aav = 0.444 is the average value, S = 0.396
is the standard deviation. By simple mathematical
transformations the approximation (11) can be
linearized. The regression becomes linear in the
estimated parameters bi. The resulting features Aline
of the linearized regression satisfy the homogeneity
condition: τmax = 1.29 < τmin = 1.41 < τ0.05cr(N) =
3.12; the Kolmogorov-Smirnov normality test: d =
0.188, λ = 1.26 < λ0.95cr = 1.36. In this case,
statistical tests of linear regression can be used to
assess the significance of the regression. The
following statistics were obtained for the linearized
regression equation:
N = 45; R = 0.82 ± 0.07, R > R0.05cr(N – 2) = 0.295
[11]; for relatively small samples (N ≥ 10) the
statistical significance of the correlation coefficient
is determined by the inequality [11]: t = 0.5ln[(1 +
R)/(1 – R)]∙(N – 3)0.5 = 7.50 > t0.05cr(N – 2) = 1.68;
the minimum sample size sufficient for the
reliability of the correlation coefficient [18]: N0.05min
= 6: RMSE = 1.977, c = 4.71∙10-4, b0 = -7.68 ± 1.70,
b1 = 1.89 ± 0.20, t(b1) = 9.30 > |t(b0)| = 4.53 >
t0.05cr(N – 2) = 2.017; F = 86.52 >> F0.05cr(f1 = 1; f2
= 43) = 4.06; sum of squared residuals: Σ0 = 168.1.
(13)
Since the inequality t(b1) > t1-αcr takes place for a
two-sided critical region, the regression coefficient
b1 is statistically significant at a significance of 1– α,
reliably greater than zero, and reflects a positive
relationship between the molecular factor Δε and the
bioresponse. Estimation of significance of the
coefficient of determination can be obtained using F
- statistics: F = R2∙(N – m – 1)/(1 – R2), which is
compared to the table value. Here m = 1 is the
number of explanatory variables. Since F >> Fcr, we
can conclude that the coefficient of determination R2
is significantly different from zero. The population
statistics Δε for the entire sample (N = 45) will be as
follows:
N = 45; ∆εav = 8.25 ± 0.22; reliability of the average
value: t = 37.5 > t0.05cr(N – 2) = 2.017; 95%
confidence interval: (7.81-8.69); ∆εmin = 4.08, ∆εmax
= 10.7; Sε = 1.417, τmax = 1.67 < τmin = 2.84 <
τ0.05cr(N) = 3.12; the Wilk-Shapiro normality test: W
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= 0.923 ≈ W0.05cr(N) = 0.926; the Pearson normality
test: χ2 = 8.87 < χ0.052,cr(df = 12) = 21.026; the
Romanowsky's normality test: |χ2 – df|/(2df)0.5 =
0.64 < 3.0; the David-Hartley-Pearson normality
test: U10.05cr(N) = 3.83 < U = (∆εmax – ∆ε min)/Sε
= 4.5 < U20.05cr(N) = 5.35; V = (17.2 ± 1.81)%; P =
2.56%; Nrepr = 36; Θ = 37.6 > 3. (14)
Here df = n – l – 1 is the number of freedom
degrees, n is the number of intervals into which the
range of variation of the random variable is divided,
l is the number of estimated distribution parameters.
It follows from inequalities (14) that the population
of elements Δε is homogeneous and has a
distribution close to the normal distribution.
According to the Chaddock scale [19], the
correlation coefficient R (13) is in the range of
values, which characterizes the relationship between
the explanatory variable and the resultant variable as
"close". Thus, at a significance level of 5%, we can
recognize the existence of a close relationship
between the events. The correlation field that
determines the distribution of observations (Fig. 1)
indicates the existence of a relationship between the
value of the energy interval Δε and the radio-
protective activity of molecules.
Fig. 1. The scattering diagram of the radio-
protective action of a number of substituted
aminothiols and their analogues (Table 1) depending
on the magnitude of the difference in electronic
energies Δε. The solid line is determined by
regression (11).
In a fairly narrow range of energies 8eV ≤ ∆ε ≤
9eV (group of chemical compounds Nos.16-24)
there is a significant change in the radio-protective
properties of low molecular weight compounds. The
energy interval ∆ε can be compared with the
threshold processes of deexcitation of metastable
states of biomacromolecules, associated both with
the interception of migrating electronic excitation in
the biosystem, as well as with the prevention of
possible molecular conformational transitions. The
threshold action of radioprotectors is important not
only when the molecular descriptor Δε is used as an
explanatory feature. In the case when ∆ε < 8.5 eV,
i.e., less than the threshold value, most of the
aminothiols analyzed here have a significant
prophylactic effect. If the value of ∆ε noticeably
exceeds the energy range of 8.5 - 9.0 eV, then the
chemical compounds of a number of substituted
aminothiols, as a rule, are weakly active in the
antiradiation action.
From experiments [20,21] aimed at studying the
radioprotective effect of cysteamine on mammalian
cells, it is known that radio-protector molecules
quickly penetrate into cells and reach nuclear DNA
without difficulties associated with transport. Low-
molecular compounds can interact with DNA [22]
and, thus, contribute to the stabilization of the
macromolecule structure by participating in the
dissipation of the excitation electronic energy into
the conformational energy of the impurity nuclear
subsystem. It is possible that conformational
transitions induced by interaction with low-
molecular compounds lead to changes in the
electronic state of the active groups of
biomacromolecules and their mutual arrangement.
This, in turn, can lead to pre-irradiation blocking of
DNA replication. It is possible that molecular
processes associated with the phenomenon of
conformational selection take place [23]. Possibly,
the instability of the initial conformation of low-
molecular chemical compounds with respect to
conformational transitions also has a preventive
effect, and the probability of transition to another
conformation is higher, the smaller the energy
interval Δε [23].
Analyzing the molecular data from Table 1, one
can notice that an increase in the energy interval Δε
is accompanied by a decrease in the dipole moment
of the molecule. This qualitative conclusion can be
verified using the Abbe-Linniсk test (13). After
ranking the data by the value of the feature Δε, the
following inequalities for dipole moments were
obtained q = 0.531 < q0.05cr(N = 45) = 0.7603. It
follows that for the sample presented in Table 1,
there is a significant trend with a statistical
significance of 0.95.
Next, let's check whether the relationship
between the variables μ and Δε is linear or non-
linear. In accordance with the physical concepts of
the participation of polar molecules in
intermolecular interactions [24], the dependence of
the energy on the value of the dipole moment must
be quadratic. Indeed, the value of the energy interval
Δε (in eV) is significantly related to the value of the
square of the total dipole moment μ2 of the
4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
, eV
A/100%
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molecule. The regression equation can be linearized
by replacing the explanatory variable μ2 with M:
Δε(M) = a0 + a1∙M, (15)
N = 45; rμ2 = - 0.74 ± 0.07, |rμ2| > r0.05cr(N – 2) =
0.310; statistical significance of the correlation
coefficient: t = |rμ2|(N – 2)0.5/(1 – rμ22)0.5 = 7.2 >
t0.05cr(N – 2) = 2.017 (two-sided critical region); the
minimum sample size sufficient for the reliability of
the correlation coefficient: N0.05min = 7; RMSE =
0.998; a0 = 9.62 ± 0.24, a1 = - 0.19 ± 0.03, t(a0) =
40.1 > |t(a1)| = 7.25 > t0.05cr(N – 2) = 2.017; sum
of squares residuals: Σ0 = 42,8.
The significance of the coefficient of
determination is determined using F-statistics: F =
r2∙(N – 2)/(1 – r2) = 52.08 >> F0.05cr(f1 = 1; f2 = 43) =
4.06. For comparison, here is also the correlation
coefficient for the regression: Δε(μ) = b0 + b1∙μ, rμ
= - 0.66.
Dipole moment statistics of the molecules:
N = 45, μav = 2.54 ± 0.14; reliability of the average
value: t = 18.1 > t0.05cr(N) = 2.014; 95% confidence
interval: (2.25-2.83); μmin = 0.91, μmax = 23.9; Sμ =
4.89; τmin = 1.69 < τmax = 2.45 < τ0.05cr(N) = 3.12;
the David-Hartley-Pearson normality test:
U10.05cr(N) = 3.75 < U = [(μmax – μmin)/Sμ] = 4.13 <
U20.05cr(N) = 5.26; Nrepr = 36, P = 6.7%; Θ = 17.6 >
3.0. (16)
The populations μ (16) and Δε (14) are
homogeneous, and the distribution of their elements
is close to the normal distribution.
The presence of a large dipole moment of the
molecule contributes to the long-range (on the
molecular scale) electrostatic interaction, which can
lead to the emergence of a microgradient and
concentration of the drug in the local area of the
biophase. In addition, the presence of a dipole
moment in a molecule suggests that such molecules
should have the property of hydrophilicity.
However, the simultaneous presence of groups of
CH2 atoms in molecules gives molecules the
opposite property - hydrophobicity. The
hydrophobicity of the molecule increases with an
increase in the number of CH2 groups. In this case,
the molecules from Table 1 refer to amphiphilic
molecules containing both hydrophobic and
hydrophilic sites from groups of atoms. As is
known, the existence of polar and nonpolar parts of
a molecule promotes the aggregation of low-
molecular chemical compounds with the formation
of molecular clusters, including those with
biological molecules.
It was shown [24] that the contribution to the
interaction energy, which is proportional to the
square of the dipole moment of the molecule, is due
to the polarization properties of the target
biosubstrate and the dipole-dipole interaction. The
higher the induction electron polarizability of an
object, the stronger its interaction with a low-
molecular compound. It is well known that if some
molecule has a constant dipole moment, then this
dipole moment causes a shift of charges in the
neighboring molecule, i.e. there is an induced dipole
moment. This results in an attraction between the
molecules due to constant and induced dipole
moments. Such interactions were called orientation-
induction interactions. In the dipole approximation,
the interaction energy is proportional to μ2 and R-6;
here R is the distance between the centers of gravity
of the molecules. It should also be noted that the
polarization properties of electronically excited
molecular systems are significantly higher
compared to the polarization of molecules in their
ground electronic state. The importance of
orientation-induction interactions for the activation
of bioactivity of molecules suggests that the region
with which the molecule interacts should have high
polarization properties. The presence of a large
dipole moment of the molecule in general indicates
the hydrophilicity of the molecule. Indeed, for the
majority of active (A ≥ 60%) in radio-protective
chemical compounds (Table 1) dipole moment on
average exceeds the dipole moment of ineffective in
radio-protective chemical compounds. Let us check
whether the average values of the dipole moment of
chemical compounds differ significantly for the
activity regions A1 (N1 = 15), A2 (N2 = 9), and A3 (N3
= 21). The following statistics have been obtained:
N1 = 15, μ1av = 3.22 ± 0.34; reliability of the average
value: t = 9.5 > t0.05cr(N1) = 2.131; 95% confidence
interval: (2.49-3.95); μ1min = 0.91, μ1max = 4.89, Sμ1 =
1.315, τmax = 1.26 < τmin = 1.76 < τ0.05cr,2(N1) = 2.497
< τ0.05cr,1(N1) = 2.617; the Wilk-Shapiro normality
test: W = 0.915 > W0.05cr(N1) = 0.881; the David-
Hartley-Pearson normality test: U10.05cr(N1) = 2.97
≈ U = [(μ1max – μ1min)/Sμ1] = 3.03 < U20.05cr(N1) =
4.170; P = 10.5%; Nrepr = 12, Θ = 9.5. (17)
N2 = 9, μ2av = 2.30 ± 0.15; reliability of the average
value: t = 15.3 > t0.05cr(N2) = 2.262; 95% confidence
interval: (1.95-2.65); μ2min = 1.18, μ2max = 2.16, Sμ2
= 0.463, τmin = 1.18 < τmax = 2.16 < τ0.05cr,2(N2) =
2.237 < τ0.05cr,1(N2) = 2.392; the Wilk-Shapiro
normality test: W = 0.873 > W0.05cr(N2) = 0.829, the
David-Hartley-Pearson normality test: U10.05cr(N2) =
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2.590 < U = [(μ2max – μ2min)/Sμ2] = 3.35 < U20.05cr(N2)
= 3.552; P = 6.7%; Nrepr = 7, Θ = 15.3. (18)
N3 = 21, μ3av = 2.15 ± 0.10; reliability of the average
value: t = 21.5 > t0.05cr(N3) = 2.086; 95% confidence
interval: (1.93-2.36); μ3min = 1.09, μ3max = 2.69, Sμ3 =
0.471, τmin = 1.15 < τmax = 2.246 < τ0.05cr,2(N3) =
2.644 < τ0.05cr,1(N3) = 2.75; the Wilk-Shapiro
normality test: W = 0.888 ≈ W0.05cr(N3) = 0.908, the
David-Hartley-Pearson normality test: U10.05cr(N3) =
3.180 < U = [(μ3max – μ3min)/Sμ3] = 3.27 < U20.05cr(N3)
= 4.490; P = 4.8%; Nrepr = 17, Θ = 20.9. (19)
It follows that the populations are homogeneous
and have a distribution close to the normal
distribution. Let us check the significance of the
difference between the average values of μ1av and
μ2av. Let us first find the ratio of the larger sample
variance to the smaller sample variance:
F1,2 = Sμ12/Sμ22 = 8.1 >
F0.05cr(f1 = N1 – 1; f2 = N2 – 1) = 3.23. (20)
The F1,2 value exceeds the tabulated value, so the
variances should be considered different at a
significance level of α = 0.05. Since inequality (20)
is satisfied, to determine the significance of the
difference in the average values, we can use the
approximate following relation (Cochran-Cox test
[12,25]):
tSμ2 = t0.05cr(f1 = N1 - 1)Sμ12 + t0.05cr(f2 = N2 - 1)Sμ22,
Sμ = [Sμ12/N1 + Sμ22/N2]0.5, (21)
t = | μ1av - μ2av| = 0.92 > Tav = tSμ2/Sμ = 0.66.
A one-sided significance test is applied here. It
follows from inequality (21) that at the significance
level α = 0.05, the average values of the dipole
moment μ differ significantly for regions A1 and A2.
Then, using relations [11,12] compare the
average values of μ2av and μ3av for regions A2 and A3:
F2,3 = Sμ32/Sμ22 = 1.03 >
F0.05cr(f2 = N2 - 1; f3 = N3 - 1) = 2.45,
t = | μ2av - μ3av| = 0.15 < tav = t0.05cr(f = N2,3 - 2)×
{N2,3∙S2,32/[N2∙N3∙(N2,3 - 2)]}0.5 = 0.94,
N2,3 = N2 + N3, S2,32 = (N2 - 1)∙Sμ22 + (N3 - 1)∙Sμ32.
(22)
Consequently, for weakly active and inactive
chemical compounds, the average values of the
populations μ2 and μ3 at the 5% significance level do
not differ significantly and we can accept the null
hypothesis, that is, they belong to the same set. In
this case, the samples μ2 and μ3 can be combined.
The following statistics for the combined population
were obtained:
N2+3 = 30; μ2+3av = 2.19 ± 0.09; reliability of the
average value: t = 24.3 > t0.05cr(N2+3) = 2.042; 95%
confidence interval: (2.02-2.37); μ2+3min = 1.09,
μ2+3max = 2.85, S2+3 = 0.463, τmin = 1.42 < τmax = 2.38
< τ0.05cr(N2+3) = 2.96; the Wilk-Shapiro normality
test: W = 0.895 ≈ W0.05cr(N2+3) = 0.927; the David-
Hartley-Pearson normality test: U10.05cr(N2+3) =
3.470 < U = [(μ2+3max – μ2+3min)/S2+3] = 3.80 <
U20.05cr(N2+3) = 4.890; P = 3.9%; N2+3repr = 24, Θ =
25.6. (23)
Statistics (23) demonstrates that the sample N2+3
= 30 is homogeneous, and the population elements
have a distribution close to normal. It is now
possible to compare the average dipole moment
values for molecules belonging to the group of
bioactive drugs (A1 ≥ 60%) and for molecules
belonging to the pooled population (A2 = 50% and
A3 ≤ 30%). Using relations (5) and (6) we obtain the
following inequalities:
F = S12/S2+32 = 8.07 >
F0.05cr(f1 = N1 – 1; f2+3 = N2+3 – 1) = 2.05,
t = | μ1av – μ2+3av| = 1.03 > Tav = 0.75. (24)
Therefore, the average values of the dipole
moments of molecules for bioactive and inactive
drugs at the 95% confidence level differ
significantly. Thus, an increase in the sample size
(up to N2+3 = 30) containing weakly active drugs and
inactive chemical compounds does not change the
significance of the difference between the average
values (21) and (24). Consequently, we can
recognize that the distinction for the average values
is not random.
Let us also check the hypothesis of the existence
of a relationship between the value of the anti-
radiation activity of chemical compounds (Table 1)
and the value of the square of the dipole moment of
molecules. For this purpose we will use the method
of contingency of features. Indeed, for the majority
of radioprotective active chemical compounds (A1 ≥
60%), the following inequality holds for the square
of the dipole moment: μ2 > μ2,av = 7.35D2. At the
same time, for weakly active and inactive drugs, the
following inequality is more likely: μ2 < μ2,av =
7.35D2. The average value of the dipole moment
square μ2,av will be taken as its threshold value.
Using the data of Table 1, we will compile a 3×2
contingency table (Table 3), which presents the
relative frequencies qij of the appearance of features
in the i-th row and j-th column, as well as
theoretically expected frequencies qij', determined
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Volume 2, 2022
by the formula (1). To test the hypothesis of the
significance of the relationship between the feature
μ2 and the radioprotective efficacy of substituted
aminothiols and their analogues, we use the chi-
square test (2): χ2 = 20.69 > χ0.052,cr(f = 2) = 5.99.
It follows from this inequality that the null-
hypothesis should be rejected and the existence with
a probability of 0.95 of a significant relationship
between the molecular feature μ2 and the
radioprotective activity of chemical compounds
should be recognized.
Table 3
Interrelation of the radioprotective action of
substituted aminothiols and their analogs with the
electronic parameter μ2
A, %
Sign μ2
≥ 7.35D2
< 7.35D2
Total
≥ 60
q11 = 9
q11' = 3
q12 = 6
q12' = 12
q1 = 15
p1 = 0.333
q1’ = 15
= 50
q21 = 1
q21' = 1.8
q22 = 8
q22' = 7.2
q2 = 9
p2 = 0.200
q2’ = 9
< 50
q31 = 0
q31’ =
4.20
q23 = 21
q23’ = 16.8
q3 = 21
p3 = 0.467
q3’ = 21
Q1 = 10
P1 =
0.222
Q2 = 35
P2 = 0.778
N = 45
3
1ii
P
=
3
1jj
p
=1.00
Apparently, the value of the square of the dipole
moment of the molecule influences the
radioprotective effect of the drug. The strength of
the bond is characterized by the contingency
coefficients (2) and (3):
K = [χ2/( χ2 + N)]0.5/Kmax = 0.74,
ϕ= [χ2 /(d – 1)/N)]0.5 = 0.68. (25)
The correction Kmax depends on the number of
rows and the number of columns of the 3×2
contingency table and can be quantified from the
following ratio [8]:
Kmax = 0.5[((v – 1)/v)0.5 + ((w – 1)/w)0.5] = 0.762.
(26)
Here w = 3 is the number of rows and v = 2 is the
number of columns, respectively; d is the smaller of
the two numbers v and w. Both contingency
coefficients (26) indicate a fairly strong relationship
between the value of the molecular trait μ2 and the
resulting trait. This conclusion does not allow
rejecting the initial assumption about the importance
of the accumulation of low-molecular compounds in
the body and their participation in intermolecular
bonds through dipole-dipole and orientation-
induction interactions in the radioprotective effect.
Let us check the significance of the relationship
between the radioprotective properties of drugs and
the square of the dipole moment of the molecule
using the following regression equation:
A(μ2)/100 = 1/[1+c∙exp(b0 + b1∙μ2)] , (27)
linearized regression statistics:
N = 45; R = - 0.60 ± 0.12, |R| > R0.05cr(N – 2) =
0.310; the minimum sample size sufficient for the
reliability of the correlation coefficient: N0.05min =
11; RMSE = 2.76, c = 4.71∙10-4, b0 = 10.41 ± 0.66,
b1 = - 0.34 ± 0.07, t(b0) = 15.68 > |t(b1)| = 4.85
> t0.05cr(N – 2) = 2.017; F = 23.52 > F0.05cr(f1 = 1; f2
= 43) = 4.06; Σ0 = 327.2 (28)
In applied statistics, an approximate rule has
been established: if the absolute value of the
correlation coefficient R exceeds the average error
of the coefficient by at least three times, then we can
reliably assume that the relationship between the
signs is not random. Used in statistics (28), the
tabular values of the Student and Fisher tests
indicate the significance of the relationship between
the value of the attribute μ2 and the radioprotective
activity of chemical compounds in Table 1.
However, as further analysis showed, the joint
consideration of the explanatory variables Δε and μ2
in the regression did not lead to a significant
improvement in the regression. That is, the inclusion
of an additional explanatory variable μ2 in the
regression (11) does not have a significant effect on
the resulting variable. At the same time, an
additional check of the relationship (15) of
molecular features Δε and μ2 showed that for
bioactive drugs (A1 ≥ 60%) there is a significant
linear relationship:
Δε(μ2) = a0 + a1μ2, N1 = 15, r1 = -0.69 ± 0.15,
adjusted correlation coefficient [11]: |r1*| = 0.71 >
r0.05cr(N1 – 2) = 0.514; estimation of the significance
of the correlation coefficient, taking into account the
Hotelling corrections [11]: uH = 0.794 > u0.05(N1) =
0.523; the minimum sample size sufficient for the
reliability of the correlation coefficient: N0.05min = 8;
RMSE = 0.955; a0 = 7.92 ± 0.45, a1 = - 0.11 ± 0.03,
t(a0) = 17.63 > |t(a1)| = 3.43 > t0.05cr(N1 – 2); F =
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11.75 > F0.05cr(f1 = 1; f2 = 13) = 4.67; straightness
sign [18]: K = [N∙(1 – R2)]0.5 = 2.80 < Kthr = 3.00.
(29)
To further test, the null-hypothesis of correlation
coefficient insignificance we will use the following
inequality, which applies to samples with a volume
≈ 10 [11]:
t = 0.5∙ln[(1 + r1)/(1 – r1)]∙(N1 – 3)0.5 =
2.94 > t0.05cr(N1 – 2) = 2.16. (30)
This inequality rejects the null-hypothesis at the
significance level α = 0.05. Consequently, we can
agree that the correlation coefficient (29) is
significant. The relationship of explanatory features
Δε and μ2 can lead to their collinarity. To check for
collinearity between variables, we use the Farrar-
Glauber test [26]:
χ2 = - [N1 – 1 – (2m + 5)/6]∙ln(1 – r12) =
8.30 > χ0.052,cr (f = 1) = 3.841. (31)
The number of explanatory variables m =1. Since
χ2 > χ2,cr, the hypothesis about the presence of
collinearity does not contradict the original data.
According to the Cheddock scale [19], for the
adjusted correlation coefficient r1* (29), the
relationship is characterized as "close connection”.
An increase in the feature μ2 is associated with a
decrease (a1 < 0) in the value of the energy interval
Δε. However, for the areas of bioactivity A2 and A3,
there is no significant relationship between the
molecular features Δε and μ2. Indeed, the correlation
coefficients at a significance level of 5% are
noticeably lower than the admissible critical values
|r2| = 0.17 < r0.05cr(f = 7) = 0.666 [11] and |r3| = 0.36
< r0.05cr(f = 19) = 0.433, respectively. It is important
to note that the relationships of molecular features
for bioactive chemical compounds and inactive (or
weakly active) drugs differ significantly. That is,
there is a structural shift in the relationships of
molecular features. This structural shift can be taken
into account as an additional, qualitative property of
chemical compounds that separates molecules that
are bioactive in terms of radioprotection from
inactive or weakly active drugs.
Dipole electrostatic forces can lead to a
significant change in the distribution of positive ions
inside the cell [27,28]. This, in turn, is reflected in
the mechanism of DNA replication exposed to
intense radiation. In addition, the electrostatic field
component of the dipole (vector value), directed
along the double helix chain, strongly polarizes the
electrons of nucleotide bases, especially in
electronically excited states, which also affects
DNA replication. Intense external irradiation can
sensitize this process, while the action of the
electrostatic dipole field stabilizes it.
Dipole-dipole or dipole-induction interactions
(both interactions are proportional to μ2) can also
determine the direction of movement of a low-
molecular compound to the activation center in the
biosystem (for example, to DNA, RNA), as well as
the binding to it. The dynamic equilibrium between
the bound state of the impurity with the target
biosubstrate and the disconnected state of the
molecules is determined by anisotropic short-range
interaction forces, of which the forces responsible
for the formation of charge transfer complexes are
the most effective. For homologous series of
compounds, the ability to complex formation
depends significantly on the energy parameter εun of
the acceptor [29], and the electron-acceptor
properties of molecules are the stronger, the lower
the molecular level εun lies on the energy scale.
We group the preparations in Table 1 according
to the value of the sign εun into two groups, prone to
complex formation (εun < 0) and inactive in this
respect (εun > 0). Using the method of conjugation of
qualitative features, we determine the statistical
characteristics of the relationship between the
radioprotective activity of preparations and their
ability to form complexes with charge transfer. In
this case, the feature εun is a dichotomous feature
that can take only two qualitative values - either
negative or positive. The significance of the
relationship between features is established using
the chi-square test. Using the numerical values qij
and qij', presented in the 3×2 contingency table
(Table 4), using relations (2) and (25), we determine
the statistics of the relationship between the
molecular trait εun and the biological response
(Aexp,%): χ2 = 9.71 > χ0.052,cr(f = (v – 1)(w – 1)) =
5.99, ϕ = 0.465, K = 0.553. These results indicate a
significant relationship between the radioprotective
activity of drugs and the position on the energy
scale of the electronic level εun in isolated
molecules.
Let us compare the average values of εunav (in
eV) for the activity regions A1, A2, and A3. The
following statistics were obtained:
N1 = 15, εun1av = -1.77 ± 0.47; 95% confidence
interval: (-2.77, -0.77); εun1min = -4.70, εun1max =
0.19, Sun1 = 1.81, τmax = 1.08 < τmin = 1.62 <
τ0.05cr,2(N1) = 2.493 < τ0.05cr,1(N1) = 2.617; the Wilk-
Shapiro normality test: W = 0.860 ≈ W0.05cr(N1) =
0.881; the David-Hartley-Pearson normality test:
U10.05cr(N1) = 2.970 ≈ U = [(εun1max – εun1min)/Sun1] =
2.71 = U20.05cr(N1) = 4.170, (32)
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Table 4
Relationship between the radioprotective activity of
substituted aminothiols and their analogues and the
electronic attribute εun
A, %
Sign εun (in eV)
Negative
Positive
Total
≥ 60
q11 = 14
q11'= 9.33
q12 = 1
q12' = 5.67
q1 = 15
p1 = 0.33
q1’ = 13
= 50
q21 = 7
q21'= 5.60
q22 = 4
q22'= 3.40
q2 = 11
p2 = 0.20
q2’ = 11
< 50
q31 = 9
q31' = 13.07
q23 = 12
q23' = 7.93
q3 = 21
p3 = 0.467
q3’ = 21
Q1 = 28
P1 = 0.62
Q2 = 17
P2 = 0.38
N = 45
3
1ii
P
=
3
1jj
p
=1.00
N2 = 9, εun2av = -0.22 ± 0.03; 95% confidence
interval: (-0.70, 0.26); εun2min = -1.43, εun2max = 0.50,
Sun2 = 0.627, τmax = 1.15 < τmin = 1.93 < τ0.05cr,2(N2)
= 2.237< τ0.05cr,1(N1) = 2.392; the Wilk-Shapiro
normality test: W = 0.919 > W0.05кр(N2) = 0.829, the
David-Hartley-Pearson normality test: U10.05cr(N2) =
2.590 < U = [(εun2max – εun2min)/Sun2] = 3.07 =
U20.05cr(N2) = 3.552, (33)
N3 = 21, εun3av = 0.23 ± 0.03; 95% confidence
interval: (-0.05, 0.50); εun3min = -0.75, εun3max = 1.78,
Sun3 = 0.611, τmin = 1.60 < τmax = 2.55 < τ0.05cr(N3) =
2.80; the Wilk-Shapiro normality test: W = 0.819 <
W0.05cr(N3) = 0.908; the David-Hartley-Pearson
normality test: U10.05cr(N3) = 3.18 < U = [(εun3max –
εun3min)/Sun3] = 4.14 < U20.05cr(N3) = 4.490, (34)
These populations are homogeneous and have a
distribution of elements close to the normal
distribution. Using (5), (6), (8), (9) and the results of
statistics (32) - (34), we check the significance of
the difference between the average values of εun1av,
εun2av, and εun3av. The following inequalities were
obtained: F1,2 = Sun12/Sun22 = 8.32 >
F0.05cr(f1 = N1 – 1; f2 = N2 – 1) = 3.23,
t = | εun1av – εun2av| = 1.55 > Tav = 0.907, (35)
F2,3 = Sun22/Sun32 = 1.05 <
F0.05cr(f2 = N2 – 1; f3 = N3 – 1) = 3.52,
t = | εun2av – εun3av| = 0.44 > tav = 0.417. (36)
Obviously, the average values of εun1av and εun2av
differ significantly from each other. Considering
inequality (36) we can admit that average values of
εun1av and εun3av are also significantly different. Thus,
the average values of εunav for the three bioactivity
regions A1, A2, and A3 differ significantly from each
other at the 95% confidence level, and the following
sequence of inequalities is observed: εun3av > εun2av >
εun1av. For bioactive radioprotectors (A1 ≥ 60%), the
molecular features εun1 are grouped around εun1av = -
1.77 eV, while for weakly active or inactive
chemical compounds, the energy values εun are most
likely localized in the region (-0.22, 0.23) eV.
At one time, Bacq and Alexander [30] propose a
hypothesis according to which one of the possible
mechanisms of radioprotection is due to the fact that
the radioprotector molecule can neutralize radicals.
As is known, the primary products of radiolysis are
electrons, free radicals, and excited molecules. For
the first selected area of the compounds in Table 1,
the energy level εun1 is negative (with the exception
of drug No.2). That is, the transfer of an electron
from a radical or other reaction center to this orbital
can become energetically favorable. For example, it
is known that under the action of radiation in the
aquatic environment, chemically highly active and
very mobile (diffusion coefficient 4.96∙10-5 cm2/s)
hydrated electrons (eaq) are formed [31,32]. For
compounds of the first group (Nos. 1, 5, 6, 9, 10),
the energy of the unoccupied one-electron level εun1
lies on the energy scale below the main energy level
of a hydrated electron, which is equal to -2.82 eV
[33]. The electron hydration time is ≈ 0.24 ps. The
hydrated electron is a very powerful reducing agent,
and the addition reaction proceeds at a high rate
[31,32]. Consequently, it is energetically favorable
for the electron eaq to move from the hydrated state
to the lower molecular level of the radioprotector.
The high polarizability of –SH and –SS– groups can
become a site of attack by a hydrated electron [32].
It is noted in the literature that the rate constant of
the reaction of a hydrated electron with effective
radioprotectors (for example, cysteine, cysteamine,
or cystamine) is even higher than with oxygen. In
[34] it is noted that self-trapped electrons can cause
a break in the carbon-sulfur bond in organic
compounds. In this case, the radioprotector
molecule can act as a neutralizer of the active
radical. At the same time, the participation of the
radioprotector molecule in other possible
mechanisms of radioprotection is also not denied
here. For most of the compounds from Table 1 that
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do not have effective radioprotection, the values of
εun are either positive or small (in absolute value)
negative values. In this case, energy is required to
attach a hydrated electron.
It is not excluded that molecules of radio-
protectors can participate both in the processes of
"healing" of damages in the DNA structure, caused
by radiation exposure [35] and participate in the
interaction with the target molecule prior to
irradiation. In the latter case, such binding
(including disulfide bonds) leads to a change in the
organism's response to the subsequent action of
radiation [3]. It is possible that the adsorption of low
molecular compounds in the biosystem before
irradiation increases the activity of biological
processes - DNA replication, synthesis of RNA,
proteins, etc., which in turn increases the body's
resistance to radiation. Moreover, the radioprotector
molecule must have such a spatial configuration that
allows the molecule to adapt to local areas of the
biophase. For example, cysteine
(HSCH2CH(NH2)COOH) is known to have
radioprotective properties, whereas its optical
stereoisomer iso-cysteine has no protective
properties, although the one-electron molecular-
orbital energies of these molecules are virtually the
same. That is, the spatial correspondence
(complementarity) between the functional groups of
the protector molecules and the biological object is
important. The emerging steric hindrances can limit
the donor-acceptor properties of molecules, since
effective binding of molecules occurs with a strong
overlap of electron shells - the maximum overlap of
the electron-donor orbital with the electron-acceptor
orbital.
Two enantiomeric forms of the same molecule
often have different biological activities. This is
because receptors, enzymes, antibodies, and other
elements of the body also have chirality, and the
structural mismatch between these elements and
chiral molecules prevents their interaction. A
similar situation occurs, for example, with regard to
the effect on blood pressure of the enantiomers, the
left-handed isomer of adrenaline compared to the
right-handed isomer. These two molecules differ
only in the spatial arrangement of the structural
elements of the molecule, which is reflected in the
interaction of the molecule with the adrenoreceptor.
Structural correspondence or mismatch between
drug molecules and chiral molecules of the biophase
turned out to be a decisive factor for the different
manifestations of the biological activity of drugs
[36]. In this case, a significant role is played by the
vector quantity - the dipole moment of the molecule,
as well as the hydrophobic - hydrophilic regions of
the radioprotector molecules.
It can be assumed that in the process of
implementation in the body of the protective
properties of a radioprotector, an important role is
played by the ability of the drug to participate in the
formation of complexes with charge transfer.
Thereby, perhaps, a temporary inhibition of
biochemical processes is carried out. It is known
[37] that the intermolecular forces that determine
the formation of complexes with charge transfer are
small (compared to covalent bonds), but,
nevertheless, they have a significant effect on the
conformational transitions of macromolecules in a
polar dielectric medium. It is known [37] that the
intermolecular forces that determine the formation
of complexes with charge transfer are small
(compared to covalent bonds), but, nevertheless,
they have a significant effect on the conformational
transitions of macromolecules in a polar dielectric
medium. From the point of view of the
manifestation of radioprotective action by drugs, the
donor properties of radioprotectors have been
repeatedly discussed in the literature [38–40].
Damage repair in this case is related both to the
actual process of intermolecular electron transfer to
the ionized bioobject, which leads to "healing" of
the damage, and to the possible formation of
complexes with charge transfer.
The donor properties of chemical compounds
generally depend on many electronic and steric
properties of the interacting molecules. However,
for the homologous series of chemical compounds,
the electronic processes of electron transfer are
significantly related to the position on the energy
scale of the highest filled εoc molecular orbital. It is
well known that the higher on the energy scale is the
MO level of energy εoc, the stronger are the donor
properties of the molecule. Let's check whether
there is a statistically significant relationship
between the anti-radiation protection of the
preparations and the position of the one-electron
MO of the energy level relative to the threshold
value εocthr. For the sample presented in Table 1 (N =
45), as a threshold (boundary) value εocthr, we take
the average value εocav = -8.78 eV (95% confidence
interval is: (-8.60, -8.96) eV); the Wilk-Shapiro
normality test: W = 0.958 > W0.05cr(N) = 0.945.
Further, we again use the statistical method of
contingencies. Let's make a 3×2 contingency table
(Table 5). Using the results presented in Table 5, as
well as formulas (2), (23) and (24), one can obtain
statistics on the relationship between the radio-
protective effectiveness of substituted aminothiols
and their analogues and the position on the energy
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scale of the one-electron MO level εocav: χ2 = 10.95 >
χ0.052,cr(f = 2) = 5.99, ϕ = 0.49, K = 0.581.
Sufficiently high statistical characteristics indicate
the existence of a significant relationship between
the radioprotective activity of molecules and their
donor properties. It is important to emphasize the
physicochemical meaning of the energy parameters
εoc and εun, which determine the level of the redox
potential of the molecule
Table 5
Relationship between the radioprotective action of
molecules and the electronic parameter εoc
A, %
Sign εoc (in eV)
| εoc | ≤ 8.78
| εoc | > 8.78
Total
≥ 60
q11 = 11
q11' = 6.0
q12 = 4
q12' = 9.0
q1 = 15
p1 = 0.33
q1’ = 15
= 50
q21 = 3
q21' = 3.6
q22 = 6
q22' = 5.4
q2 = 9
p2 = 0.244
q2’ = 9
< 50
q31 = 4
q31' = 8.4
q23 = 17
q23' = 12.6
q3 = 21
p3 = 0.47
q3’ = 21
Q1 = 18
P1 = 0.400
Q2 = 27
P2 = 0.600
N = 45
3
1ii
P
=
3
1jj
p
=1.00
Let's check the significance of the difference
between the average values εoc1av, εoc2av
и εoc3av.
Using relations (5), (6), (8), (9) the following
inequalities were obtained:
F1,2 = (Soc1/ Soc2)2 = 8.93 >
F0.05cr(f1 = N1 – 1; f2 = N2 – 1) = 3.23,
t = |εoc1av – εoc2av| = 0.400 > Tav
= 0.365, (37)
F3,2 = (Soc3/ Soc2)2 = 2.85 <
F0.05cr(f3 = N3 – 1; f2 = N2 – 1) = 3.15,
t = |εoc2av – εoc3av| = 0.24 < tav
= 0.26. (38)
The average values of εoc1av for highly active
chemical compounds differ significantly, at a
confidence level of 0.95, from the values of εoc2av
and εoc3av for weakly active or inactive drugs (37).
At the same time, for chemical compounds for
which the bioactivity A2 = 50% or A3 < 50%, the
energies of the highest occupied orbital do not differ
statistically significantly (38).
Population statistics εoc1 (in eV):
A1 ≥ 60%, N1 = 15, εoc1av= -8.41 ± 0.19; 95%
confidence interval: (-8.81, -8.00), εoc1min = -9.57,
εoc1max = -7.38; Soc1 = 0.738; τmax = 1.40 < τmin = 1.57
< τ0.05cr,2(N1) = 2.493 < τ0.05cr,1(N1) = 2.617; the Wilk-
Shapiro normality test: W = 0.925 > W0.05cr(N1) =
0.881; |V| = 8.8%; P = 2.3%; N1repr = 12, |Θ| = 44.1,
(39)
A2 = 50%, N2 = 9, εoc2av = -8.80 ± 0.08; 95%
confidence interval: (-8.99, -9.27), εoc2min = -9.27,
εoc2max = -8.38; Soc2 = 0.247; τmax = 1.72 < τmin =
1.89 < τ0.05cr,2(N2) = 2.237 < τ0.05cr,1(N2) = 2.392; the
Wilk-Shapiro normality test: W = 0.940 > W0.05cr(N2)
= 0.829; |V| = 2.8%; P = 0.9%; N2repr = 8, |Θ| = 5.28,
(40)
A3 < 50%, N3 = 21, εoc3av = -9.04 ± 0.09; 95%
confidence interval: (-9.23, -8.85), εoc3min = -9.85,
εoc3max = -8.85; Soc3 = 0.417; τmax = 0.46 < τmin = 1.95
< τ0.05cr,2(N3) = 2.644 < τ0.05cr,1(N3) = 2.750; the Wilk-
Shapiro normality test: W = 0.943 > W0.05cr(N3) =
0.908; |V| = 4.6%; P = 1.0%; N3repr = 19, |Θ| = 19.6.
(41)
Sets εoc (39) - (41) are homogeneous and have a
distribution close to the normal distribution.
The performed statistical analysis showed that
the radioprotective effectiveness of a number of
mercaptoethylamine derivatives and their analogs
depends on the different electronic properties of the
molecules. As it turned out, there are some threshold
values for all discussed electronic characteristics of
samples from Table 1, and going beyond these
values leads to a significant change in the preventive
properties of drugs. This result allows us to use the
methods of multivariate regression analysis and by
analogy with equation (11) we can write the
following nonlinear regression equation:
A/100 = 1/[1+c∙exp(b0 + b1∙εoc + b2∙εun +
b3 ∙Δε + b4∙μ2)]. (42)
For the regression parameter c the value obtained
for the regression (11) was taken. Regressions of
this type are usually called combined forms of
regression. It is important to note that in regression
(42) the explanatory variables Δε and μ2 as well as
εun and ∆ε are closely related. For example, the
relationship between the molecular features εun and
∆ε is as follows: N = 45, r = 0.92. In applied
statistical analysis, it is roughly accepted [8] that if
the value of the pairwise correlation coefficient |r| >
0.8, then the explanatory variables are collinear.
Below are the detailed statistics of the relationship.
At the same time, there is practically no relationship
between the explanatory variables εun and εoc: r =
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0.16. In general, the presence of a paired linear
relationship between several explanatory variables
is defined as multicollinearity. Multicollinearity
between variables can lead to a decrease in the
accuracy of regression estimation and even to the
impossibility of assessing the influence of
explanatory variables on the resulting attribute [8].
It is known, that if one of the explanatory variables
can be represented as a linear combination of other
explanatory variables, then the system of normal
equations may not have a unique solution.
Therefore, the variable ∆ε can be excluded from the
regression equation. As a result, we obtain the
following three-factor regression equation:
A/100 = 1/[1+c∙exp(b0 + b1∙εoc + b2∙εun + b3∙μ2)]. (43)
After linearizing equation (43), the following
multiple regression statistics were obtained:
N = 45, multiple correlation coefficient: R1 = 0.870
> R0.05cr(f1= m;f2 = ν) = 0.415 [41], R12 = 0.757,
adjusted coefficient of determination [11]: R1*2 =
0.74, RMSE = 1.745; c = 4.71∙10-4, b0 = -19.664 ±
4.022, b1 = -3.306 ± 0.455, b2 = 1.406 ± 0.305, b3 =
-0.104 ± 0.075; |t(b1)| = 7.26 > |t(b0)| = 4.88 > t(b2)
> 4.61 > t0.05cr(f = N - m - 1) = 2.021 > |t(b3)| = 1.39;
the significance of the coefficient of multiple
determination: F = 41.86 > F0.05cr(f1 = m;f2 = N - m -
1) = 2.83; Σ = 124.85; AIC = 1.1537, SC = 1.3588,
SS = 0.2660. (44)
Here m = 3 is the number of explanatory
variables; ν = N - m - 1; R1-α(m;ν) is the multiple
correlation coefficient. The residuals of regression
(43) are normally distributed. Kolmogorov -
Smirnov normality test for regression residuals: dmax
= 0.1022, λ = dmax∙N0.5 = 0.686 < λ0.8cr = 1.07. The
Wilk-Shapiro normality test is also performed: W =
0.951 > W0.05cr(N = 45) = 0.945. Σ is the sum of
squares of the regression residuals. Standardized
(normalized) regression coefficients bi* are defined
as follows [42]:
b1*= b1Sεoc /SAct = - 0.570 ± 0.079,
b2*= b2Sεun/SAct = - 0.599 ± 0.130,
b3* = b3Sμ2/SAct = - 0.179 ± 0.129. (45)
Here, the index Act ≡ A/100% (after linearization
of the regression equation); Sεoc = 0.586, Sεun =
1.447, Sμ2 = 5.87, SAct = 2.234. Standardized
coefficients make it possible to compare
quantitatively the influence of each explanatory
variable on the variability of the resulting attribute.
Using the standardized coefficients (45), one can
determine the approximate coefficient of
determination, which in an additive form allows one
to make estimates of the relative contributions of
each explanatory variable to the variability of the
resulting attribute:
Rappr2 = b1*∙rεoc,Act + b2*∙rεun,Act + b3*∙rμ2,Act
= 0.263 + 0.378 + 0.116 = 0.757. (46)
The approximate coefficient of determination (46)
coincides with the coefficient of determination of
the regression (44). Here rεoc,Act = -0.456, rεun,Act =
0.648, rμ2,Act = -0.594 are paired correlation
coefficients between the explanatory variables and
the resulting feature (after transformation to a linear
form). All correlation coefficients in absolute value
are greater than the permissible table value r0.05cr(f =
N – 2) = 0.300 [11]. From relation (46) it follows
that the maximum contribution to the explanation of
the variability of bioactivity comes from the
electronic energies εun (26.3%) and εoc (37.8%). The
contribution from the dipole moment of the
molecule is much lower and amounts to only 11.6%.
Statistics (44) also provides information criterion
Akaike [43] relative quality of a linear statistical
model for a given data set. The information criterion
is defined as follows:
AIC = 2m/N + ln(Σ/N). (47)
Here m is the number of explanatory variables; Σ
is the sum of the squares of the regression residuals;
N is the number of observations. The AIC test
establishes a trade-off between the magnitude of the
residual sum of squares and the number of
explanatory variables. The first term is the penalty
for using additional variables, the second term is the
penalty for large variance. As the number of
variables in a linear model increases, the first term
in (47) increases and the second term decreases,
because usually increasing the number of variables
in a regression reduces the residual sum of squares.
Regression residuals should be normally distributed.
The equation (47) usually also includes a constant
value of 1 + ln(2π), which is omitted here because it
is not essential for comparison tests. The Akaike test
quantifies the relative amount of information that is
lost when building a statistical model. The less
information is lost (that is, the smaller the AIC value
(47)), the higher the quality of the model. When
comparing statistical models, preference is given to
the model for which the AIC test is the smallest, that
is, the model that minimizes information loss. The
test is useful only when comparing linear statistical
models, and the size of the compared samples N
must be the same. Recently, the Schwarz criterion
has also been frequently used [44]:
SC = (m + 1)ln(N)/N + ln(Σ/N). (48)
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The Schwartz criterion is similar to the Akaike
criterion, but uses more stringent penalty functions.
Practical applications of the (47) and (48) tests have
shown that the Schwartz test is somewhat more
reliable than the Akaike test in relative comparison
of statistical models. The estimate obtained using
this indicator is considered consistent. For a
comparative assessment of the quality of models,
one can also use the ratio: SS = Σ0.5/(N – m). This
ratio is usually closely related [45] to the test values
of Akaike and Schwartz.
Thus, approximately 24% of the variance
remains unexplained, which can be attributed to
unaccounted for factors or due to random variation
in the raw data. The adjusted coefficient of
determination is determined as follows [8,11]:
R*2 = 1 – (1 – R2)(N – 1)/(N – m – 1). (49)
Obviously, the adjusted coefficient of
determination (49) depends on the number of
explanatory variables in the regression. The adjusted
coefficient of determination is used for the purpose
of comparing models with different numbers of
factors, so that the number of explanatory variables
does not affect the R2 statistics. The multiple
correlation coefficient is determined from the
following relationship:
R1-α(f1= m;f2 = ν) = [m∙F1-α(m,ν)/(ν + m∙F1-α(m,ν)]0.05.
(50)
Here ν = N – m – 1; F1-α is 100∙(1 – α)% quantile of
distribution F(m;ν). The null-hypothesis H0: R = 0 is
rejected at the α significance level, since R > R1-αcr.
At a given significance level α = 0.05, the multiple
correlation coefficient is much larger than the
critical value and, therefore, its difference from zero
is not accidental. In applied statistics, it is accepted
that if the coefficient of determination R2 > 0.75,
then the relationship between the effective feature
and the explanatory variables can be characterized
as strong. The significance of the correlation
coefficient can also be checked using the t-criterion
(15): t = R1(N – m – 1) 0.5/(1 – R12)0.5 = 11.3 >
t0.05cr(f = 41) = 2.021. (51)
According to inequality (51) we can admit that
the model adequately describes the relationship
between the resultant variable and the explanatory
variables. Signs εoc and εun can be defined as
intensive indicators (to a lesser extent this applies to
μ2), which are directly related to cause-and-effect
relationships between bioactivity and the structure
of molecules. The statistics of the sampling sets εoc
and εun, (the statistics of the set μ2 are given in (16))
and the resulting feature A*act (values after
linearization are used) will be as follows:
εocav = -8.79 ± 0.09; N = 45; 95% confidence
interval: (-8.96, -8.60), εocmin = -9.85, εocmax = -7.38;
Soc = 0.59; τmin = 1.81 < τmax = 2.41 < τ0.05cr(N) =
3.12; the David-Hartley-Pearson normality test:
U10.05cr(N) = 3.75 < U = [(εocmax – εocmin)/Soc] = 4.19
< U20.05cr(N) = 5.26;
εunav = -0.53 ± 0.22; N = 45; 95% confidence
interval: (-0.96, -0.09), εunmin = -4.70, εunmax = 1.78;
Sun = 1.45; τmax = 1.59 < τmin = 2.88 < τ0.05cr(N) =
3.12; the David-Hartley-Pearson normality test:
U10.05cr(N) = 3.75 < U = [(εunmax – εunmin)/Sun] = 4.47
< U20.05cr(N) = 5.26;
Aact*av = 7.51 ± 0.51; N = 45; 95% confidence
interval: (6.50 - 8.53), Aact*min = 2.70, Aact*max = 11.9;
SА = 3.40; τmax = 1.29 < τmin = 1.42< τ0.05cr(N) = 3.12;
the Kolmogorov-Smirnov normality test: dmax =
1.26, λ = dmax∙N0.5 = 1.26 < λ0.95cr = 1.36; the David-
Hartley-Pearson normality test: U10.05cr(N) = 3.75 >
U = [(Aact*max – Aact*min)/SA] = 2.71 < U20.05cr(N) =
5.26. (52)
The negative value of the coefficient b1 means
that with increasing energy εoc (negative value), the
radioprotective properties of the preparations
increase. At the same time, a decrease in the εun
level on the MO energy scale is accompanied by a
decrease in the antiradiation activity of drugs (b2 >
0). An additional check showed that the explanatory
factors εun and μ2 are closely related. The correlation
coefficient is |r23| = 0.786 > R0.05cr(f = N - 2) = 0.300.
For the remaining explanatory variables the
following pairwise correlations were obtained: r12 =
0.162 и r13 = 0.118. The regression residuals (43)
are normally distributed (the Wilk-Shapiro test: W =
0.957 > W0.05cr(N = 45) = 0.925). The presence of
multicollinearity of the variables is tested with the
Farrar-Glauber test:
χ2 = - (N – 1 – (2m + 5)/6)∙ln
3,32,31,3
3,22,21,2
3,12,11,1
det
rrr rrr rrr
= 45.8 > χ0.05cr(f = m(m – 1)) = 12.592. (53)
The collinearity of the explanatory variables is
also indicated by the value:
t23 = |r23|∙(N – m)0.5/(1 – r232)0.5 =
8.24 > t0.05cr(f = N – m) = 2.02. (54)
Since inequalities (53) and (54) are satisfied, the
hypothesis of multicollinearity does not contradict
the original data. One of the highly correlated
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explanatory variables must be eliminated from the
regression equation. Which of the variables should
be removed is determined as follows. The least
significant of all regression coefficients is the
coefficient b3 (44). Next, the values tij =rij∙(N –
m)0.5/(1 – rij)0.5 (i = 1,2,3 and j = 1,2,3), are
calculated. The index for t23 has the maximum
value: t23 = 8.24 > t0.05cr(f = N – 3) = 2.02 > t12 =
1.08 > t13 = 0.7. Therefore, the third explanatory
variable μ2 can be excluded from the regression
(44). Thus, the regression equation (44) can be
replaced by a two-factor equation:
A/100 = 1/[1+c∙exp(b0 + b1∙εoc + b2∙εun)] . (55)
After linearizing the regression equation, the
following statistics were obtained:
N = 45, the multiple correlation coefficient is equal:
R2 = 0.86 > R0.05cr(f1 = 2; f2 = 42) = 0.365 [41], R22 =
0.740, R2*2 = 0.740, RMSE = 1.764; the significance
of the coefficient of multiple determination: F =
60.48 > F0.05cr(f1 = m; f2 = N – m – 1) = 3.22; c =
6.79∙10-4, b0 = 20.362 ± 4.034, b1 = -3.319 ± 0.46, b2
= 1.744 ± 0.18; t(b2) > 9.36 > |t(b1)| = 7.29 >|t(b0)| =
5.05 > t0.05cr(f = 42) = 2.02; Σ1 = 130.73 is the sum
of squares of residuals; the test of normality of the
population of residuals: W = 0.951 > W0.05cr(N = 45)
= 0.945; the Kolmogorov-Smirnov normality test
for the residuals: dmax = 1.102, λ = 0.6855 < λ0.2cr =
1.07; the regression quality tests: AIC = 1.1554, SC
= 1.3203, SS = 0.2659; b1* = b1∙Soc/SA = -0.573 ±
0.079, b2* = b2∙Sun/SA = 0.749 ± 0.079. (56)
Reducing the number of variables in regression
(55) compared to regression (44) preserves the
quality of the regression. Moreover, the Schwarz
test (SC) indicates an improvement in the quality of
the regression. The following estimates of the
contribution of each explanatory variable to the
variability of the resultant variable were obtained:
Rappr2 = b1*∙rεoc,Act + b2*∙rεun,Act =
0.263 + 0.481 = 0.744. (57)
The approximate coefficient of determination
(57) is very close to the coefficient of determination
(56). Thus, the molecular parameters εoc and εun
actually determine the pharmacodynamic stage of
drug action. Regression (55) does not contradict
regression (43). As mentioned above, the pair
correlation coefficient between the explanatory
variables εv and εun is insignificant: r1,2 = 0.16.
Therefore, it can be recognized that there is no
collinearity between the explanatory variables.
Since the regression residuals (55) are normally
distributed (W = 0.951 > W0.05cr(N) = 0.945), the
collinearity of the explanatory variables can be
quantified using the Farrar-Glauber relation:
χ2 = - [N – 1 – (2m + 5)/6]∙ln
2,22,1
1,21,1
det rr rr
=
1.10 < χ0.052,cr(f = 1) = 3.841. (58)
Since inequality (58) is satisfied, we can agree that
there is no significant collinearity between the
variables at the 95% confidence level.
It is known [5] that elongation of the
hydrocarbon chain in the NH2(CH2)kSH molecule
for k = 2, 3, 4 leads to a decrease in the
radioprotective effect of the chemical compound.
Indeed, using the data of Table 1, it can be seen that
for preparations Nos. 19 and 29 (k = 3 and 4), the
electron-acceptor ability of these compounds
noticeably weakens compared to preparation No. 14
(k = 2), the hydrophobic contribution is increased
and at the same time the energy Δε increases. In
accordance with equations (42) and (43), these
changes can lead to a decrease in radiation
protection. In addition, for the chain of carbon-
hydrogen atoms (CH2)k there is a change in the
effective charges of carbon atoms (positive values).
The effective charge of an atom characterizes the
shift of the electron density along the chemical bond
and this is a quantitative measure of the polarization
of the chemical bond. The greater the change, the
farther the carbon atom is located from the acceptor.
Such electron density distribution leads to the
appearance of centers with different reactivity. It is
possible that the different biological activity of α-
homocysteine and β-homocysteine is related to this.
The replacement of the amine group in
compound No.14 by a methyl group (No. 35) or by
an isoelectronic (in terms of the number of electrons
on the outer shell) hydroxyl group (No. 30) changes
the electronic properties of the molecules so that it
leads to a decrease in their antiradiation action and
simultaneously reduces the donor-acceptor
properties of the molecules as a whole. The electron
affinity (A, eV) values are known for some
substituents [46]. It is well known, that the measure
of the electron affinity of an atom, molecule, or
group of atoms is the amount of energy released
when an electron is attached to it. A comparative
analysis of the observed electron affinity values for
the substituents in the R1 position of NH2 (A = 0.74
eV), CH3 (A = 1.05-1.08 eV), N(CH3)2 (A = 1.08
eV), NHCH3 (A = 1.56 eV) and OH (A = 1.83 ±
0.04 eV) demonstrates that this sequence is
associated with a decrease in the radio-protective
effect of the drugs (Nos. 14, 35, 31, 30, 36): 60 (70),
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10, 10(40), 10(50), and 0%(0%), respectively. The
brackets indicate the radioprotection given in [47].
Therefore, the R1CH2CH2R2 molecule in this case is
asymmetric in terms of the energy parameter, that is,
in terms of the electron affinity (A) of the
substituents. The molecule is, as it were, “polarized”
(R2 = SH, A = 2.32±0.01 eV [46]) by its ability to
accept or donate an electron. The greater this energy
"asymmetry", the higher the radioprotective effect
of the chemical compound.
In the process of irradiation in a living organism,
a hydroxyl radical (OH∙) arises, which is extremely
chemically active and destroys almost any molecule
it encounters. Acting on SH-groups, histidine and
other amino acid residues of proteins, hydroxyl OH∙
causes denaturation of the latter and inactivates
enzymes. In this case, the radioprotector molecule
containing the SH group can intercept the hydroxyl
molecule. Since the SH group has a high electron
affinity, electron transfer from the radical to the
radioprotector molecule is possible. In nucleic acids,
the OH∙ radical destroys carbohydrate bridges
between nucleotides and, thus, breaks DNA and
RNA chains, resulting in mutations and cell death.
In addition, the decay of the negative molecular ion
produces the H∙ ion, which has a very high kinetic
energy [48]. Apparently, the presence of non-protein
SH groups in the molecule is a necessary condition
for the effectiveness of low molecular weight
aminothiols, but not sufficient. It was established
[4] that there is a connection between the protective
effect of radioprotectors and the concentration of
SH-groups in body tissues.
It can also be noted that the groups of R1 atoms
have a relatively low electron affinity, but a high
ionization potential, which is noticeably higher than
that of the SH substituent (I = 10.5 eV). For
example, the ionization potentials of the NH2 and
OH groups are known [46] to be 11.4 and 13.18 eV,
respectively. It is possible that the function of the R1
substituents is to orient the radioprotector molecule
in space (for example, due to intermolecular
hydrogen bonding) in such a way that the SH
substituent is available for interaction with radicals,
and the NH2 group is oriented in such a way as to
participate in the formation of the NH+∙∙∙N hydrogen
bond. There are experimental confirmations [49]
that in real biological systems there is a hydrogen
bond NH+∙∙∙N. A radioprotector molecule can
participate in the formation of such a specific bond,
given its specific spatial arrangement. This
assumption is supported by the fact that the
symmetric SHCH2CH2SH molecule (meaning
symmetry in terms of the electron affinity energy A)
exhibits no radio-protective effect [50], although the
electron affinity values of both acceptor substituents
R1 = SH and R2 = SH are the highest of the
substituents presented here. A similar situation
exists for S,S-ethyldiisothiuronium and S,S-
propyldiisothiuronium molecules. Both of these
compounds do not have effective radioprotection
[51]. The same characteristic changes are revealed
upon passing from compound No. 2 to compound
No. 41, when the substituent NH2 (in position R1)
changes to the isoelectronic group of OH atoms.
Replacement of the hydrogen atom (electron affinity
of the hydrogen atom A = 0.77 eV [46]) at the
amine group NH2 in chemical compound No.14
with the group of atoms H2C=CHCH2 (electron
affinity A ≈ 0.1 - 2.1 eV [46] (comparison of
various data; a semi-empirical quantum-chemical
calculation gives a value of A = 1.0 eV.)) or per
group of CH3 atoms (electron affinity A = 1.05–1.08
eV [46]) is also accompanied by a decrease in
survival (preparations No. 23 and No.36). Molecule
No.34 can also be added to this scheme.
Replacement of the SH substituent (No. 10) with the
isoelectronic OH substituent (No.34), with a lower
electron affinity energy, also leads to a decrease in
radioprotective activity. In this series of compounds,
it is β-mercaptoethylamine (cystamine, becaptan,
mercamin) that has the best radioprotective
properties. Obviously, substituents through chemical
bonds affect the electronic distribution in the entire
molecule. Addition of the acceptor substituent to the
hydrocarbon chain shifts the electron density along
the chain of σ-bonds of carbon atoms toward the
acceptor, and the greater the shift, the further away
from the acceptor the carbon atom is. This, in turn,
is accompanied not only by shifts in the electron
density in covalent chemical bonds, but also by the
energy of molecules, which inevitably affects the
physical and chemical properties of the drug.
The lack of radioprotective effect of iso-cysteine
(an isomer of cysteine) compared to cysteine may be
due to the conformational properties of the
molecule. The distance between the groups of SH
and NH2 atoms for iso-cysteine varies so much in
three-dimensional space that this does not allow
their donor-acceptor properties to manifest
themselves. Iso-cysteine does not form mixed
disulfides with proteins, but, like radiosensitizers,
binds to them by other intermolecular bonds [3].
The substitution of the thiol group SH (electron
affinity is 2.32 eV) for the iso-electronic (according
to the number of valence electrons) OH group
(No.34; electron affinity is 1.83 eV) in chemical
compound No.14 also noticeably reduces the
complexation activity of this compound. Comparing
preparations No.30, No.35 with preparations No.14
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and No.34, it can be noted that the highest
radioprotective activity is achieved if the R1
substituent in the R1CH2CH2R2 molecule has a
relatively low electron affinity (for example, NH2: A
= 0.74 eV; I = 11.4 eV), while the R2 substituent
(for example, SH: A = 2.32 eV; I = 10.4 eV) has a
noticeably higher value than the R1 substituent. A
decrease in the affinity energy for the R2 substituent
is accompanied by a decrease in the protection
effect. For example, for molecules No.14 (SH
substituent, A = 2.32 eV), No.16 (SCN substituent,
A = 2.17 eV) and No.25 (SCH3 substituent, A ≈ (2.0
– 2.5) eV) there is the following radioprotection
sequence: 60, 50 and 30%. Moreover, the change in
the value of the ionization potential I of substituents
has the opposite direction to the change in the value
of electron affinity.
It is important to note that among the chemical
compounds of Table 1, it is the SH substituent that
has the highest (experimentally observed) value of
electron affinity among the substituents used here.
However, this is only one side of the properties of
the molecules associated with the manifestation of
the radioprotective effect of the drugs. A large
positive value of εun (1.72 eV) for molecule No.34
indicates that this chemical compound is practically
incapable of forming intermolecular complexes due
to donor-acceptor interactions. In addition, the large
value of the molecular parameter ∆ε (10.7 eV)
apparently also completely excludes the possibility
of the molecule's participation in electron-
conformation transitions. The blocking of the SH
and NH2 functional groups (Nos. 25, 26, 28, 31)
violates the threshold conditions established above
for the energy parameters of the molecules, which
correlates with a decrease in the radioprotective
effect. The substitution of a hydrogen atom in the
SH group is also accompanied by a decrease in the
electron affinity of the substituent, which is
associated with a change in the radioprotective
activity of the drugs: No.14 (R2 = SH, A = 2.32 eV,
A = 60%); No.16 (R2 = SCN, A = 2.17 eV, A =
50%); No.26 (R2 = SCH2CH3, A = 1.18 eV, A =
20%). For comparison, the electron affinity of the
sulfur atom is 2.077 eV. If the addition of other
atoms to the sulfur atom, for example, drugs No.14
and No.16, lead to an increase in the electron
affinity of the substituent compared to the affinity of
the sulfur atom, then, as follows from Table 1, the
radioprotective activity of the drug is noticeable. At
the same time, the addition of a group of CH2CH3
atoms to the sulfur atom (No.26) reduces the
substituent's electron affinity to such an extent that it
becomes less than the atomic value for sulfur. The
hydrophobic properties of the molecule also
increase. For such a substituent, this is accompanied
by a noticeable decrease in the antiradiation activity
of the chemical compound. In addition, the
substitution of the hydrogen atom at the amine and
thiol groups creates steric hindrances that hinder the
participation of these compounds in the processes of
electron transfer, intermolecular approach, and
conformational selection. In particular, for example,
the formation of complexes with charge transfer is
most effective at such distances between the
reagents when there is a significant overlap of the
interacting molecular orbitals.
Comparison of the influence of changes in the
factors of equation (55) on the variability of
radioprotective activity of sulfur-containing amino
acids: cysteine (No.17) and iso-cysteine (No.27), the
first of which has pronounced antiradiation
protection, is of some interest. Without denying the
possibility of the participation of cysteine in the
defense of the body through other possible
mechanisms of protection against intense radiation
[6], which are not discussed here, the following
circumstance should be noted. Moving the carboxyl
group from the α-position to the β-position with
respect to the mercapto group unfavorably changes
the important energy parameter of the molecule εun,
and the estimates of bioactivities in this case differ
by more than a factor of two using regression (55).
The SH and CH groups can participate in the
formation of an intermolecular hydrogen bond, and
the bond strength is characterized by the following
sequence: OH > NH > SH > CH. The SH substituent
belongs to the classical proton donors with
participation in the formation of a hydrogen bond.
As the hydrogen bonding energy increases, the
redistribution of electron density affects all of the
atoms of the molecules that make up the molecular
complex, which can ultimately lead to profound
changes in the physical and chemical properties of
substances.
3.2 The relationship of information and
electronic features of molecules
It was shown [52] that the information molecular
character dH1 = pH∙log2pH - pC∙log2pC is related to
the biological activity of a chemical compound.
Here, the pH and pC probabilities determine the
proportion of hydrogen and carbon atoms in the
molecule. The total molecular information function
for a discrete set of atoms is quantified as follows
[45]: H = - Σipilog2pi, pi = ni/N, ni is the number of
atoms of sort i; N is the total number of atoms in the
molecule. The summation is performed over all
sorts of atoms in the molecule.An analysis of the
interrelations of molecular factors showed that the
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information function dH1 is related to the value of
the electronic energy εun. The following regression
was obtained for bioactive chemical compounds
(Nos.1-15) (radioprotective activity is equal to A1 ≥
60%):
εun(dH1)1 = a01 + a11dH11, N1 = 15, R1 = 0.86 ±
0.07, R1* = 0.87 > R0.05cr(N1 – 2) = 0.514; estimation
of the significance of the correlation coefficient,
taking into account Hotelling's corrections [11]: uH
= 1.214 > u0.05(N1) = 0.523; the minimum sample
size sufficient for the reliability of the correlation
coefficient: N0.05min = 5; RMSE(S1) = 0.951, a01 = -
0.92 ± 0.28, a11 = 21.65 ± 3.53, t(a11) = 6.14 >
|t(a01)| = 3.25 > t0.05cr(N1 – 2) = 2.16; F = 37.67 >
F0.05cr(f1 = 1; f2 = 13) = 4.67; sum of the residuals
squares: Σ1 = 11.75; the Wilk-Shapiro normality test
for the residuals: W = 0.962 > W0.05cr(N1) = 0.881;
straightness sign: K = 1.97 < Kthr = 3.00 [18]. (59)
Similarly, we write a linear regression for a
sample containing chemical compounds Nos.16-25
(the bioactivity is equal to A2 = 50%):
εun(dH1)2 = a02 + a12dH12, N2 = 9, R2 = 0.82 ± 0.12,
R2* = 0.84 > R0.05cr(N2 – 2) = 0.666; assessment of
the significance of the correlation coefficient, taking
into account the Hotelling corrections: uH = 1.037 >
u0.05(N2) = 0.693; the minimum sample size
sufficient for the reliability of the correlation
coefficient: N0.05min = 6; RMSE(S2) = 0.383, a02 = -
0.60 ± 0.16, a12 = 13.69 ± 3.61, t(a12) = 3.80 >
|t(a02)| = 3.70 > t0.05cr(N2 – 2) = 2.365; F = 14.4 >
F0.05cr(f1 = 1; f2 = 7) = 5.59; sum of the residuals
squares: Σ2 = 1.027; the Wilk-Shapiro normality test
for the residuals: W = 0.954 > W0.05cr(N2) = 0.829;
straightness sign: K = 1.72 < Kthr = 3.00. (60)
For small sample sizes N ≤ 15 the best estimate
[11] of the correlation coefficient is R* = R∙[1 +
0.5(1 - R2)/(N - 3)]. Let's check whether the two
regressions (59) and (60) can be combined into one
regression, i.e. the same relationship of signs for
these samples or different. To do this, we use the
Chow test [53]. We first obtain the regression for
the combined sample, i.e. including the populations
A1 and A2:
εun(dH11+2) = a0 + a1dH11+2, N = 24, R = 0.89 ±
0.04, R* = 0.90 > R0.05cr(N - 2) = 0.404; the
minimum sample size sufficient for the reliability of
the correlation coefficient: N0.05min = 5; RMSE(S) =
0.783, a0 = -0.89 ± 0.16, a1 = 21.69 ± 2.37, t(a1) =
8.93 > |t(a0)| = 5.45 > t0.05cr(N – 2) = 2.074; F = 79.7
> F0.05cr(f1 = 1; f2 = 22) = 4.30; sum of the residuals
squares: Σ = 13.50, normality of residual
distribution (Wilk-Shapiro test): W = 0.966 >
W0.05cr(N) = 0.918; straightness sign: K = 2.14 < Kthr
= 3.00. (61)
Population statistics of εun:
N = 24; εunav = -1.19 ± 0.34; 95% confidence
interval: (-1.88, -0.49), εunmin = -4.70, εunmax = 0.50;
Sun = 1.647; τmax = 1.03 < τmin = 2.13 < τ0.05cr,2(N) =
2.701 < τ0.05cr,1(N) = 2.800; the Pearson normality
test: χ2 = 3.81 < χ0.052,cr(df = 12) = 21.026, the
David-Hartley-Pearson normality test: U10.05cr(N) =
3.25 ≈ U = [(εunmax – εunmin)/Sнс] = 3.16 < U20.05cr(N)
= 4.60; Nrepr = 19; (62)
population statistics of dH11+2:
N = 24, dH11+2av = -0.014 ± 0.014; 95% confidence
interval (-0.043, 0.015), dH11+2min = -0.147,
dH11+2max = 0.079; SdH11+2 = 0.069; τmax = 1.35 < τmin
= 1.93 < τ0.05cr,2(N) = 2.701 < τ0.05cr,1(N) = 2.800; the
Pearson normality test: χ2 = 0.83 < χ0.052,cr(df = 13) =
22.362; the David-Hartley-Pearson normality test:
U10.05cr(N) = 3.25 < U = [(dH11+2max –
dH11+2min)/SdH11+2] = 3.28 < U20.05cr(N) = 4.60; Nrepr
= 19. (63)
The Chow test is defined by the following
inequality:
F = [(Σ – Σ 1 – Σ 2)∙(N – 2m – 2)] ×
[( Σ 1 + Σ 2)∙(m + 1)]-1 = 0.566 <
F0.05cr(f1 = m + 1;f2 = N – 2m – 2) = 3.49. (64)
Inequality (64) indicates, first, that the two
regressions can be combined into a single
regression, and, second, that there is no structural
shift in the relationship between the energy εun and
the attribute dH1. Both regressions (59) and (60) are
statistically significant. The combined regression is
of higher quality than the separate regressions (59)
and (60). According to the Cheddock scale [19], the
linear relationship between the attributes εun and
dH1 (59) and (60) is characterized as "very close".
At the same time, for the region of weak bioactivity
A3 ≤ 30%, sample volume N3 = 21 (Nos. 25-45)
there is no relationship between the signs. The linear
correlation coefficient is insignificant: R3 = 0.09 <
R0.05cr(f = 19) = 0.433; F = 0.16 << F0.05cr(f1 = 1; f2
= 19) = 4.38. In this case, the events are mutually
independent for any pair of random values εun and
dH1. Thus, there is a structural shift in the
relationship between the signs of εun and dH1 when
moving from bioactive to inactive or weakly active
drugs. It can be assumed that such a shift in the
relationships is associated with a change in the anti-
radiation activity of chemical compounds.
The statistics of populations dH11, dH12, dH13
will be as follows:
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A1 ≥ 60%: N1 = 15, dH11av= - 0.039 ± 0.019; 95%
confidence interval: (- 0.079, 0.0005), dH11min = -
0.147, dH11max = 0.066; SdH11 = 0.072, τmax = 1.46 <
τmin = 1.50 < τ0.05cr,2(N1) = 2.493 < τ0.05cr,1(N1) =
2.617; the Wilk-Shapiro normality test: W = 0.905 >
W0.05cr(N1) = 0.881; the David-Hartley-Pearson
normality test: U10.05cr(N1) = 2.970 = U =
[(dH11max – dH11min)/SdH11] = 2.96 < U20.05cr(N1) =
4.17; Nrepr = 12;
A2 = 50%: N2 = 9, dH12av
= 0.028 ± 0.013; 95%
confidence interval: (-0.001, 0.057), dH12min = -
0.024, dH12max = 0.079; SdH12 = 0.038, τmax = 1.34 <
τmin = 1.37 < τ0.05cr,2(N2) = 2.493 < τ0.05cr,1(N2) =
2.617; the Wilk-Shapiro normality test: W = 0.937 >
W0.05cr(N2) = 0.829; the David-Hartley-Pearson
normality test: U10.05cr(N2) = 2.59 < U = [(dH12max
– dH12min)/SdH12] = 2.71 < U20.05cr(N2) = 3.552; Nrepr
= 7;
A3 ≤ 30%: N3 = 21, dH13av = 0.028 ± 0.011; 95%
confidence interval: (0.006 - 0.050), dH13min = -
0.038, dH13max = 0.110; SdH13 = 0.049, τmin = 1.35 <
τmin = 1.34 < τ0.05cr(N3) = 2.64; the Wilk-Shapiro
normality test: W = 0.890 ≈ W0.05cr(N3) = 0.908; the
David-Hartley-Pearson normality test: U10.05cr(N3)
= 3.18 ≈ U = [(dH13max – dH13min)/SdH13] = 3.02 <
U20.05cr(N3) = 4.49; Nrepr = 17. (65)
Populations dH1i=1,2,3 are homogeneous and have a
distribution of elements close to normal. Let us
check the significance of the difference between the
average values of the information function dH1av for
the bioactivity areas A1, A2, and A3. For the
compared regions A1 and A2, A2 and A3, in
accordance with relations (5), (6), (8) and (9), the
following inequalities were obtained:
F1,2 = SdH112/SdH122 = 3.67 >
F0.05cr(f1 = N1 - 1; f2 = N2 - 1) = 3.52,
t = |dH11av – dH12av| = 0.065 > Tav = 0.041, (66)
F3,2 = SdH132/SdH122 = 1.69 <
F0.05cr(f3 = N3 - 1; f2 = N2 - 1) = 3.44,
t = |dH12av – dH13av| = 0.057 > tav = 0.024. (67)
Inequalities (66) and (67) indicate that the mean
values of the information function dH1 are
significantly different for regions A1 and A2 and for
regions A3 and A2. Thus, the information function
dH1, as well as the quantum molecular signatures
εun and εoc, allows us to separate bioactive drugs
from weakly active or inactive chemical
compounds. Consequently, the electronic sign ε and
the information function dH1, derived from
different representations of the molecular structure,
do not contradict each other.
It can also be shown that the regression
coefficients a11 (59) and a12 (60) differ statistically
insignificantly. Let us preliminarily check whether
the variances of the residuals differ significantly [8].
The verification is carried out using a relation that
has an F-distribution:
F = (S1/S2)2 = 2.73 <
F0.05cr(f1 = N1 – 2; f2 = N2 – 2) = 3.55. (68)
The numerator (68) has a large dispersion. This
result also does not contradict the Romanovsky test
[54]: Q = S12∙(N1 – 3)/[S22∙(N1 – 1)] = 2.05, SΞ =
{2∙(N1 + N2 – 4)/[(N2 – 1)∙(N1 – 5)]}0.5 = 0.845, Ξ =
|Q – 1|/SΞ = 1.24 < 3.0. There is a large despersion
in the numerator for Q. In this case, you can use the
following relation to estimate the difference
between the regression coefficients a11 and a12 [8]:
S2 = [(N1 – 2)∙S12 + (N2 – 2)∙S22]/(N1 + N2 – 4),
Ω12 = 1/[(N1 – 1)∙SdH112] + 1/[(N2 – 1)∙SdH122],
t = |a11 – a12|/(S2∙Ω12)0.5 = 0.75 <
t0.05cr(f = N1 + N2 – 4) = 2.08. (69)
Inequality (69) allows us to agree with the null
hypothesis that the regression coefficients a11 and
a12, which determine the slope of the lines, differ
insignificantly from each other.
Let us also compare the correlation coefficients
[8]:
Λ = |z1 – z2|∙[(N1 – 3)-1 + (N2 – 3)-1]-0.5 =
0.272 < Λ0.05cr = 1.96, (70)
here z = 0.5∙ln[(1 + R)/(1 – R)] = 1.1513∙lg[(1 +
R)/(1 – R)] is the normalizing Fisher transform [11]
for the correlation coefficient R. It follows from
inequality (70) that the correlation coefficients of
the regressions also do not differ significantly.
Using the values of z1 and z2, let us test the
hypothesis that the composite estimate (com) of the
correlation coefficient is different from zero:
zcom = [z1∙(N1 – 3) + z2∙(N2 – 3)] ×
(N1 + N2 – 6) -1 = 0.941. (71)
The test is carried out with the help of the following
ratio, which has a normal distribution:
Λ = zcom ∙[N1 + N2 – 6]0.5 = 3.99 > Λ0.05cr = 1.96.
(72)
Inequality (72) suggests that there is a significant
relationship between the molecular features εns and
dH1 for the bioactivity regions A1 and A2 at the 5%
level of significance. This result does not contradict
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the conclusion that follows from (69). Thus, it is
advisable to split the total sample into two parts
only if the decrease in variance is significantly
greater than the remaining unexplained variance
when using two regressions.
Further analysis showed that the electronic
energies εun and Δε for a number of chemical
compounds from Table 1 are very closely related to
each other:
Δε(εun) = a0 + a1εun, N = 45, R = 0.92 ± 0.02, R >
R0.05cr(N – 2); the minimum sample size sufficient
for the reliability of the correlation coefficient:
N0.05min < 5; RMSE = 0.581, a0 = 8.74 ± 0.09, a1 =
0.94 ± 0.06, t(a0) = 94.6 > t(a1) = 15.43 > t0.05cr(N
– 2) = 2.014; F = 238.1 > F0.05cr(f1 = 1; f2 = 43) =
4.08; regression residuals are normally distributed
(the Wilk-Shapiro test): W = 0.960 > W0.05cr(N) =
0.945; sum of residuals squares: Σ = 14.55;
straightness sign: K = 2.63 < Kthr = 3.00. (73)
Now let's check whether the variational series
has a structural shift when moving from bioactive
chemical compounds (region A1) to relatively
weakly bioactive drugs (A2 = 50%). The following
two regressions were obtained for bioactive
chemical compounds (Nos. 1-15):
Δε(εun)1 = a01 + a11εun, N1 = 15, R1 = 0.95 ± 0.03,
R1* = 0.96 > R0.05cr(N1 – 2) = 0.514; the minimum
sample size sufficient for the reliability of the
correlation coefficient: N0.05min < 5; RMSE1 =
0.424; a01 = 7.81 ± 0.16, a11 = 0.67 ± 0.06, t(a01) =
60.0 > t(a11) = 10.6 > t0.05cr(N1 – 2) = 2.160; F =
112.0 > F0.05cr(f1 = 1; f2 = 13) = 4.67; regression
residuals are normally distributed (the Wilk-Shapiro
test): W = 0.976 > W0.05cr(N1) = 0.881; sum of
squares of residuals: Σ1 = 2.348; straightness sign: K
= 1.21 < Kthr = 3.00, (74)
and for weak drugs (Nos. 16-24):
Δε(εun)2 = a02 + a12εun, N2 = 9, R2 = 0.92 ± 0.06, R*
= 0.93 > R0.05cr(N2 – 2) = 0.666; estimation of the
significance of the correlation coefficient, taking
into account the Hotelling corrections: uH = 1.431 >
u0.05(N2) = 0.693; the minimum sample size
sufficient for the reliability of the correlation
coefficient: N0.05min < 5; RMSE2 = 0.211; a02 = 8.75
± 0.08, a12 = 0.76 ± 0.12, t(a02) = 116.5 > t0.05cr(a12)
= 6.40 > t0.05cr(N2 – 2) = 2.365; F = 40.9 >
F0.05cr(f1 = 1; f2 = 7) = 5.59; the Wilk-Shapiro test for
residuals: W = 0.904 > W0.05cr(N2) = 0.829; sum of
squares of residuals: Σ2 = 0.313; straightness sign: K
= 1.15 < Kthr = 3.00. (75)
Linear regression for the merged population N = N1
+ N2 has the following statistics:
Δε(εun) = a0 + a1εun, N = 24, R = 0.93 ± 0.03, R >
R0.05cr(N – 2) = 0.404; the minimum sample size
sufficient for the reliability of the correlation
coefficient: N0.05min < 5; RMSE = 0.538, a0 = 8.31 ±
0.14, a1 = 0.80 ± 0.07, t(a0) = 61.0 > t(a1) = 11.73
> t0.05cr(N – 2) = 2.074; F = 137.5 > F0.05cr(f1 = 1; f2
= 22) = 4.30; the Wilk-Shapiro test for residuals: W
= 0.915 = W0.05cr(N) = 0.916; sum of squares of
residuals: Σ = 6.368; the sign of straightness: K =
1.80 < Kthr = 3.00. (76)
Population statistics Δε and εun for pooled samples:
N = 24, Δεav = 7.37 ± 0.29; 95% confidence interval:
(6.77-7.96), Δεmin = 4.08, Δεmax = 9.35; S∆1 = 1.417;
τmax = 1.40 < τmin = 2.32 < τ0.05cr,2(N) = 2.701 <
τ0.05cr,1(N) = 2.800; the Wilk-Shapiro test: W = 0.923
> W0.05cr(N) = 0.918, the Pearson normality test: χ2 =
1.03 < χ0.052,cr(df = 11) = 19.675; the David-
Hartley-Pearson normality test: U10.05cr(N) = 3.34 <
U = [(Δεmax – Δεmin)/S∆1] = 3.72 < U20.05cr(N) = 4.71;
P = 3.9%; Nрrepr = 19,
N = 24, εunav = -1.19 ± 0.34; 95% confidence
interval: (-1.88,-0.49), εunmin = -4.70, εunmax = 0.50;
Sun = 1.647; τmax = 1.03 < τmin = 2.13 < τ0.05cr,2(N) =
2.701 < τ0.05cr,1(N) = 2.800; the Wilk-Shapiro test: W
= 0.819 < W0.05cr(N) = 0.918, the Pearson normality
test: χ2 = 3.81 < χ0.052,cr(df = 12) = 21.026; the
David-Hartley-Pearson normality test: U10.05cr(N) =
3.34 ≈ U = [(εunmax – εunmin)/Sun] = 3.20 < U20.05cr(N)
= 4.71; Nрrepr = 19. (77)
Then the ratio (64) for the Chow test is calculated:
F = 13.94 > F0.05cr(f1 = m + 1; f2 = N–2m–2) = 3.49.
(78)
Thus, in accordance with inequality (78), it can
be assumed that the relationship between molecular
features Δε and εun undergoes a statistically
significant structural shift in the transition from the
A1 bioactivity region to the A2 bioactivity region.
Therefore, it is not recommended to use regression
(76) built on pooled samples to interpret the
relationship of features. The difference Σ – Σ1 – Σ2 is
an indicator of the improvement in the quality of the
model when the sample size is divided into two
parts. Thus, the null-hypothesis about the absence of
a structural shift in the sample data is rejected.
Therefore, for statistical analysis, two samples
should not be combined into one, and the transition
from region A1 to region A2 has a qualitative jump in
the relationship of molecular features εun and ∆ε.
Similarly, we check for the presence of a structural
shift for the relationship of explanatory variables for
samples from areas A2 and A3. For inactive or
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weakly bioactive chemical compounds (Nos. 25-
45), linear regression has the following statistics:
Δε(εun)3 = a03 + a13εun, N3 = 21, R3 = 0.79 ± 0.09,
R3* = 0.80 > R0.05cr(N3 – 2) = 0.433; estimation of
the significance of the correlation coefficient, taking
into account the Hotelling corrections: uH = 1.023 >
u0.05(N3) = 0.438; the minimum sample size
sufficient for the reliability of the correlation
coefficient: N0.05min = 6; RMSE = 0.417; a03 = 9.07
± 0.10, a13 = 0.86 ± 0.15, t(a03) = 93.18 > t(a12) =
5.63 > t0.05cr(N3 – 2) = 2.093; F = 31.7 > F0.05cr(f1
= 1; f2 = 19) = 4.38; the Wilk-Shapiro test for the
residuals: W = 0.970 > W0.05cr(N3) = 0.908; the sum
of the squares residuals: Σ3 = 3.305; straightness
sign: K = 2.75 < Kthr = 3.00. (79)
The statistics of the population of elements εun
for area A3 will be as follows:
N3 = 21, εunav = 0.23 ± 0.13; 95% confidence
interval: (-0.05, 0.50), εunmin = -0.75, εunmax = 1.78;
Sun = 0.611, τmin = 1.60 < τmax = 2.54 < τ0.05cr,2(N3) =
2.644 < τ0.05 cr,1(N3) = 2.750; the Wilk-Shapiro
normality test: W = 0.816 < W0.05 cr(N3) = 0.908; the
Pearson normality test: χ2 = 16.6 < χ0.052,cr(df = 15)
= 24.996; the David-Hartley-Pearson normality test:
U10.05 cr(N3) = 3.18 < U = [(εunmax – εunmin)/Sun3] =
4.14 < U20.05cr(N3) = 4.49; Nrepr = 17. (80)
For the combined sample (areas A2 and A3) the
linear regression statistics will be as follows:
Δε(εun) = a0 + a1εun, N = 30, R = 0.84 ± 0.06, R* =
0.85 > R0.05cr(N – 2) = 0.361; the minimum sample
size sufficient for the reliability of the correlation
coefficient: N0.05min = 5; RMSE = 0.385, a0 = 8.97
± 0.07, a1 = 0.90 ± 0.11, t(a0) = 126.2 > t(a1) =
8.06 > t0.05 cr(N – 2) = 2.048; F = 65.0 > F0.05cr (f1
= 1; f2 = 28) = 4.20; regression residuals are
normally distributed (the Wilk-Shapiro test): W =
0.957 > W0.05cr(N3) = 0.927; the sum of the squares
residuals: Σ = 4.158; straightness sign: K = 2.88 <
Kthr = 3.00. (81)
Using relation (69), as well as the results (73),
(76) and (81), the following inequality for the Chow
test was obtained:
F = 1.94 <F0.05cr(f1 = m +1;f2= N – 2m – 2 ) = 3.34.
(82)
Since F < Fcr, the null-hypothesis of structural
stability of the variation series at the 95%
confidence level should be accepted. Therefore,
combining samples A2 and A3 into one sample is
allowed. That is, for the relationship of molecular
features Δε and εun, there is a qualitative and
quantitative structural shift in the transition from
bioactive drugs (A1 region) to inactive or relatively
weakly active drugs (A2 and A3 regions). Thus, the
relationships between molecular features Δε and εun
for bioactive and inactive chemical compounds
differ significantly. Figure 2 clearly shows the
structural shift for the relationship of signs Δε and
εun, which separates bioactive chemical compounds
from inactive or relatively weakly active drugs.
There are also two lines of regression equations (74)
and (81). Thus, taking into account the results (64),
(74)-(76) and (79), we can assume the existence of
homogeneous or heterogeneous information arrays.
Taking into account the significant relationship
between the features Δε and εun (73), as well as
significant statistics (59) and (60), structural
changes can also be found in the relationships of the
feature Δε with the molecular features Z [52] and
dH1 (59). The molecular feature Z is associated with
the pseudopotential of the molecule [45,55].
Fig.2. Scatterplots for bioactive drugs (∆) and for
inactive or weakly active chemical compounds (•).1
- linear regression (74). 2 - linear regression (81).
The following designations are used here: DE ≡ ∆ε,
Eun ≡ εun.
It is important to check the presence of such a
relationship, since the sign Δε (or sign εun) and the
molecular signs Z and dH1 were obtained for
samples based on different physical concepts of the
molecular structure. As shown in [45], the dH1
factor correlates with the hydrophobic properties of
molecules, that is, it is associated with the
pharmacodynamic stage of drug action. Let's check
the relationship between the signs Δε and Z. For
bioactive chemical compounds (A1 ≥ 60%), the
following straight-line regression was obtained:
Δε(Z)1 = a01 + a11Z1, N1 = 15, R1 = - 0.82 ± 0.09,
|R1*| = 0.83 > R0.05cr(N1 – 2) = 0.514; estimation of
the significance of the correlation coefficient, taking
into account Hotelling's corrections: uH = 1.086 >
u0.05(N1) = 0.523; the minimum sample size
sufficient for the reliability of the correlation
6420 2
4
6
8
10
12
Eun, eV
DE, eV
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Volume 2, 2022
coefficient: N0.05min = 6; RMSE = 0.763; a01 = 14.1
± 1.48, a11 = -2.60 ± 0.51, t(a0) = 9.52 > |t(a1)| =
5.07 > t0.05cr(N1 – 2) = 2.160; F = 25.7 > F0.05cr(f1
= 1; f2 = 13) = 4.67; the Wilk-Shapiro test for
regression residuals: W = 0.887> W0.05cr(N1) = 0.881;
the sum of the residuals squares: Σ1 = 7.581; the
sign of straightness: K = 2.16 < Kthr = 3.00. (83)
The statistics of the Z1 population will be as
follows:
N1 = 15, Z1av = 2.85 ± 0.10; 95% confidence
interval: (2.63-3.07), Z1min = 2.286, Z1max = 3.60, SZ1
= 0.397, τmin = 1.42 < τmax = 1.89 < τ0.05cr,2(N1) =
2.493 < τ0.05cr,1(N1) = 2.617; the Wilk-Shapiro
normality test: W = 0.950 > W0.05cr(N1) = 0.881; V =
13.9%; P = 3.6%; Nrepr = 12; Θ = 27.8. (84)
Linear regression for the region A2 = 50%:
N2 = 9, Δε(Z)2 = a02 + a12Z2, R2 = - 0.68 ± 0.20,
|R2*| = 0.72 > R0.05cr(N2 – 2) = 0.666; estimation of
the significance of the correlation coefficient, taking
into account Hotelling's corrections: uH = 0.741 >
u0.05(N2) = 0.692; the minimum sample size
sufficient for the reliability of the correlation
coefficient: N0.05min = 8; RMSE = 0.405, a02 = 12.2
± 1.46, a12 = -1.39 ± 0.57, t(a0) = 8.32 > |t(a1)| =
2.45 > t0.05cr(N2 – 2) = 2.365; F = 6.02 > F0.05cr(f1
= 1; f2 = 7) = 5.59; the Wilk-Shapiro test for
regression residuals: W = 0.943 = W0.05cr(N2) =
0.829; the sum of the residuals squares: Σ1 = 1.150;
the sign of straightness: K = 2.08 < Kthr = 3.00.
(85)
For the population N2, the Z2 statistics will be as
follows:
N2 = 9, Z2av = 2.56 ± 0.08; 95% confidence interval
(2.37-2.76), Z2min = 2.286, Z2max = 3.00; SZ2 = 0.252;
τmin = 1.10 < τmax = 1.73 < τ0.05cr,2(N2) = 2.237 <
τ0.05cr,1(N2) = 2.392; the Wilk-Shapiro normality test:
W = 0.927 > W0.05cr(N2) = 0.829; V = 9.8%; P =
3.3%; Nrepr = 8; Θ = 30.5. (86)
Comparing linear pairwise regressions (83) and
(85) we can note the decrease in the quality of
regressions when the bioactivity of chemical
compounds decreases. Let's check if the two
regressions (83) and (85) are significantly different.
Again, we will use the Chou test (64). Let's
previously the regression for the combined sample:
Δε(Z) = a0 + a1Z, N = 24, R = - 0.79 ± 0.08, |R*| =
0.80 > R0.05cr(N – 2) = 0.404; the minimum sample
size sufficient for the reliability of the correlation
coefficient N0.05min = 6; RMSE = 0.882; a01 = 15.7 ±
1.37, a1 = - 3.02 ± 0.50, t(a0) = 11.43 > |t(a1)| = 6.1
> t0.05cr(N – 2) = 2.074; F = 37.2 > F0.05cr(f1 = 1; f2 =
22) = 4.30; the Wilk-Shapiro test for regression
residuals: W = 0.957 > W0.05cr(N) = 0.916; the sum
of the residuals squares: Σ = 17.15; the sign of
straightness: K = 2.93 < Kthr = 3.00. (87)
The statistics of the population Z will be as
follows:
N = 24, Zav = 2.74 ± 0.08; 95% confidence interval:
(2.59-2.90), Zmin = 2.286, Zmax = 3.60; SZ = 0.372;
τmin = 1.22 < τmax = 2.31 < τ0.05cr,2(N) = 2.701 <
τ0.05cr,1(N) = 2.800; the Wilk-Shapiro normality test:
W = 0.929 > W0.05cr(N) = 0.916; V = 13.6%; P =
2.8%; Nrepr = 19; Θ = 36.1. (88)
Using the statistics (83)-(87) we check the Chow
test (64):
F = 9.64 > F0.05cr(f1 = 2; f2 = 20) = 3.49. (89)
Inequality (89) allows us to reject the null-
hypothesis and admit that regressions (83) and (85)
are significantly different. This is also indicated by
the sign of straightness of the combined sample K
(87), which practically coincides with the threshold
value. The tendency to reduce the radioprotective
activity of chemical compounds (Table 1) is
accompanied by a tendency to reduce the quality of
regression equations. Moreover, for a sample
containing inactive or weakly active drugs (N3 =
21), the relationship between molecular features Δε
and Z decreases almost to zero (correlation
coefficient R = 0.03). Thus, in this case, when the
electronic attribute Z is used as an explanatory
variable, there is a structural shift for the
relationship between the factors Δε and Z during the
transition from active to inactive chemical
compounds in terms of radioprotection. The result
obtained does not contradict the statistical
conclusions (73), (76) and (80). It is important to
note that the resulting variables in (73) - (80) and
the explanatory variables εun and Z in (84) - (86)
were obtained based on completely different ideas
about the structure of the molecule. The sign εun is
determined using quantum mechanical calculations
of the electronic structure of molecules, the sign Z is
associated with the pseudopotential of the molecule,
and the signs H and dH1 are informational functions
of the molecule.
The list of chemical compounds included
compounds for which a noticeable antiradiation
protective effect could be expected, but,
nevertheless, these drugs in practice are not
effective radioprotectors. One of the possible
reasons for limiting biological activity is the
processes associated with the hydrophobic
properties of molecules. One of the possible reasons
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for limiting biological activity is the processes
associated with the hydrophobic properties of
molecules. For this reason, such chemical
compounds as, for example, Nos. 28, 29, 36, 44 may
be ineffective in terms of radioprotection. The
results presented in Table 6 indicate the existence of
a relationship between the radioprotective effect of
drugs and their molecular informational sign dH1.
There are two qualitative assessments for dH1. For
active drugs (A1 ≥ 60%), the dH1 value is
predominantly negative, while for inactive or
weakly active chemical compounds, the dH1 value
is positive. The chi-square test at a significance
level of α = 0.05, as well as the contingency
coefficients K and C, make it possible to draw a
statistically justified conclusion about the presence
of a statistical relationship between the
radioprotective effect and the value of the molecular
information sign dH1 [52]. Indeed, since χ2 > χ2,cr
(Table 6), then with a probability of 0.95 we can
accept the hypothesis of the existence of a
relationship between the resulting feature
(bioactivity) and the explanatory sign dH1.
Table 6
Relationship between the radioprotective effect of
substituted aminothiols and their analogues and the
information factor dH1.
A, %
Sign dH1, bits
The negative
The positive
Total
≥ 60
q11 = 11
q11' = 6.67
q12 = 4
q12' = 8.33
q1 = 15
p1 = 0.33
q1’ = 15
= 50
q21 = 2
q21' = 4.00
q22 = 7
q22' = 5.00
q2 = 9
p2 = 0.20
q2’ = 9
≤ 30
q31 = 7
q31' = 9.33
q23 = 14
q23' = 11.67
q3 = 21
p3 = 0.47
q3’ = 21
Q1 = 20
P1 = 0.444
Q2 = 25
P2 = 0.556
N = 45
3
1ii
P
=
3
1jj
p
=1.00
Statistics of the contingent signs
χ2 = 7.91 >χ0.052,cr(f = 2) = 5.99, ϕ = 0.419, K =
0.507
Similarly, it can be shown that the independently
determined molecular sign Z (Table 1) is also
significantly related to the energy interval Δε for
bioactive chemical compounds (region A1). A linear
regression equation was obtained, for which the
correlation coefficient turned out to be significant
and, accordingly, has the value |R*| = 0.83 >
R0.05cr(N1 – 2) = 0.514; F = 14.4 > F0.05cr(f1 = 1; f2 =
13) = 4.67. Evidence of linearity of regression: K =
2.16 < Kcr = 3.0. At the same time, for area A3
(sample size N3), there is no relationship between
features Δε and Z. The correlation coefficient is
insignificant and equals |R| = 0.04 < R0.05cr(N3 – 2) =
0.433. Thus, in this case, when using the sign Z as
an explanatory variable, there is a structural shift in
the relationship between Δε and Z during the
transition from bioactive chemical compounds to
inactive ones. Figures 3A and 3B show significant
relationships between informational molecular
features (H, dH1) and pseudopotential feature Z,
which are evaluated using different ideas about the
molecular structure.
A
B
Fig. 3. Δ – area A1 ≥ 60%; × – area A2 = 50%; • –
area A3 ≤ 30%. A. The linear regression: Z(H) = a +
b∙H, N = 45 , a = 0.46 ± 0.11, b = 13.22 ± 0.06,
RMSE = 0.101, R = 0.96 ± 0.04. B. The linear
regression: Z(dH1) = a + bdH1, N = 45, a = 2.69 ±
0.02, b = - 5.09 ± 0.26, R = - 0.95 ± 0.05, RMSE =
0.119.
Obviously, in this case, the relationship is
characterized by a homogeneous variance of the
random error of the regression model. Taking into
account the close relationship (Figures.3A and 3B)
1 1.5 2 2.5
2
2.5
3
3.5
4
H, bits
Z, conv. units
0.20.10 0.1 0.2
2
2.5
3
3.5
4
dH1, bits
Z, conv. units
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of molecular features: the factor Z, the total
information function H (associated with the
diversity of the molecular structure) and the partial
information function dH1, as well as the relationship
(83) - (86), it is possible to establish relationships
∆ε(H) and ∆ε(dH1) for areas of bioactivity A1, A2
and A3.
The initial sample did not include, for example,
such chemical compounds: mercamine ascorbate
(dose 0.59 mM/kg; protection 70%), mercamin
nicotinate (dose 0.76 mM/kg; protection 0%) or 1-
amino-3-mercaptopropane (dose 2.26 mM/kg;
protection 10%) [56]. For these compounds,
quantum chemical calculations of the electronic
structure of molecules have not been performed.
However, an approximate theoretical estimate of
their radioprotective activity can be obtained. The
information function of dH1 was calculated for
these drugs: -0.025, 0.023 and 0.066 bits. These
estimates do not contradict the results given in Table
6.
4 Conclusion
The discovered interrelations of molecular factors
with the radioprotective action of low molecular
weight aminothiols and their analogs demonstrated
that the bioactivity of drugs is complex and depends
on a combination of various molecular factors.
These factors determine the possible participation of
radioprotectors in the primary radiation
physicochemical processes occurring in the body,
increasing its radioresistance. It is important to note
that the energy factors of molecules are
characterized by some threshold values that separate
highly active radiation injury modifiers from weakly
active drugs.
The relationships established make it possible to
assess the radioprotective effectiveness of a
chemical compound of a number of substituted
aminothiols and their analogues without performing
complex and cumbersome quantum mechanical
calculations of the electronic structure of molecules.
It is essential that the information molecular
features, as well as the electronic factor Z, make it
possible to make an approximate assessment of the
bioactivity of the drug, having information only
about the gross formula of the chemical compound.
An important result is also the fact that for the
analyzed aminothiols, the quantum mechanical
parameter of the molecule Δε is statistically
significantly associated with both the Z factor and
molecular information functions, which were
obtained based on different ideas about the
molecular structure, not directly based on quantum
mechanical calculations of the electron molecular
structures. Statistically significant molecular
parameters εoc and εun characterize electronic
processes in which exogenous molecules can
participate as radioprotectors, and a significant
molecular factor μ2 determines the ability of these
molecules to accumulate in local areas of the
biophase prior to irradiation. It should also be noted
that explanatory variables, the evaluation of which
is based on the use of different and independent
representations, namely, on detailed quantum
chemical calculations, the use of partial information
functions of molecules, or the pseudopotential
method, which plays an important role in the
quantum theory of solids, are given in this case, to
comparable results.
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