The quantitative structure-property relationships for the dye affinity (DAF, kJ mol-1) of azo dyes were established using the Monte Carlo method by generating optimal SMILES-based descriptors. The index of ideality of correlation (IIC) and the correlation intensity index (CII) improved the model's predictive potential, specially when they were used simultaneously. The statistical quality of the best model on the validation set is it n = 18, R2 = 0.9468, and RMSE = 1.26 kJ mol− 1.
Research Article
Monte Carlo technique to study of the adsorption affinity of azo dyes with applying new statistical criteria of the predictive potential
https://doi.org/10.21203/rs.3.rs-1745462/v1
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Dye affinity
Azo dyes
QSPR
Computational chemistry
CORAL software
Correlation Intensity Index.
Dyes are colored compounds with affinity for a substrate to which they are applied. Dyes interact with substrates through several mechanisms depending on the physical and chemical properties of both the dye and substrate. Therefore the affinity and the nature of the interaction between a colorant and substrate define the quality of dyes. In the basis of their origin, colorants can be classified as natural or synthetic. Synthetic and natural colorants can be applied on different substrates, but artificial colorants dominate the market on amount of their availability and easy application [1]. The production and application methods of synthetic colorants pose some environmental challenges such as pollution of water bodies and occupational health issues in humans [2, 3]. The affinity of azo dyes to a substrate (DAF, kJ mol-1) is a significant technological and ecologic indicator [4].
Quantitative structure-property/activity relationships (QSPRs/QSARs) are a tool for developing models for the different endpoints. QSPR can be applied to build up models for DAF [5–10]. The model can be built up with the Monte Carlo method [11]. A QSPR model obtained from a stochastic process should be considered a random event [12]. However, the reliability of the QSPR is extremely important.
The present study aims to assess the so-called index of ideality of correlation (IIC) [13, 14] and the correlation intensity index (CII) [15, 16] as tools to determine the reliability of QSPR for DAF of azo dyes.
Data
The experimental data dye affinity (DAF, kJ mol-1) of 72 azo dyes were taken from the literature [5]. We randomly split these compounds into an active training set (≈ 25%), a passive training set (≈ 25%), a calibration set (≈ 25%), and a validation set (≈ 25%). Each of these sets has its task. The active training set serves to calculate correlation weights which give as large as possible the correlation between experimental and predicted endpoints for the active training set. The passive training set has to inspect whether these data give a good correlation coefficient for similar compounds in the passive training set. The calibration set is job to detect overtraining. The task for the validation set is the final estimation of the predictive potential of the model. Here we examine ten random splits.
Optimal SMILES-based descriptor
The optimal SMILES-based descriptor DCW(T,N) is applied for a predictive model of the endpoint in this equation:
$$DAF= {C}_{0}+{C}_{1}\times DCW(T,N)$$
1
$$DCW\left(T,N\right)=\sum _{ }^{ }CW({S}_{k})+\sum _{ }^{ }CW({SS}_{k})$$
2
T is an integer to separate SMILES attributes as rare or non-rare. The non-rare SMILES are employed to build up the model. The rare SMILES are not involved in building the model. N is the number of epochs of the optimization of the correlation weights. Sk is a SMILES atom, i.e., one symbol of the SMILES line (e.g. ‘=’, ‘O’) or a group of symbols that cannot be examined separately (e.g. ‘Cu’, ‘%11). SSk is two SMILES atoms. CW(Sk) and CW(SSk) correlate with these SMILES attributes.
The Monte Carlo optimization
Eq. 2 needs the numerical data for the correlation weights and the Monte Carlo optimization is a tool to calculate them. Here four target functions for the Monte Carlo optimization are examined:
$${TF}_{0}={r}_{AT}+{r}_{PT}-\left|{r}_{AT}-{r}_{PT}\right|\times 0.1$$
3
$${TF}_{1}={TF}_{0}+{IIC}_{C}\times 0.5$$
4
$${TF}_{2}={TF}_{0}+{CII}_{C}\times 0.3$$
5
$${TF}_{3}={TF}_{0}+{{IIC}_{C}\times 0.3+CII}_{C}\times 0.5$$
6
\({r}_{AT}\) and \({r}_{PT}\) are correlation coefficients between observed and predicted endpoints for the active and passive training sets, respectively.
IIC C is the index of ideality of correlation [13, 14]. IICC is calculated with data on the calibration set as follows:
$${IIC}_{C}={r}_{C}\frac{\text{m}\text{i}\text{n}({}_{ }{}^{-}{MAE}_{C},{}_{ }{}^{+}{MAE}_{C}) }{\text{m}\text{a}\text{x}({}_{ }{}^{-}{MAE}_{C},{}_{ }{}^{+}{MAE}_{C}) }$$
7
$$\text{min}\left(x,y\right)=\left\{\begin{array}{c}x, if x<y\\ y,otherwise\end{array}\right.$$
8
$$\text{max}\left(x,y\right)=\left\{\begin{array}{c}x, if x>y\\ y,otherwise\end{array}\right.$$
9
$${}_{ }{}^{-}M{AE}_{C}=\frac{1}{{}_{ }{}^{-}N}\sum \left|{\varDelta }_{k}\right|, {}_{ }{}^{-}N is the EquationNumber of {\varDelta }_{k}<0$$
10
$${}_{ }{}^{+}M{AE}_{C}=\frac{1}{{}_{ }{}^{+}N}\sum \left|{\varDelta }_{k}\right|, {}_{ }{}^{+}N is the EquationNumber of {\varDelta }_{k}\ge 0$$
11
$${\varDelta }_{k}={observed}_{k}-{calculated}_{k}$$
12
The observed and calculated are corresponding values of the endpoint.
The correlation intensity index (CII), similarly to the above IIC, was developed to improve the quality of the Monte Carlo optimization used to build up QSPR/QSAR models.
CII is calculated as follows [15, 16]:
$${CII}_{C}=1-\sum {Protest}_{k}$$
13
$${Protest}_{k}=\left\{\begin{array}{cc}{R}_{k}^{2}-{R}^{2},& if {R}_{k}^{2}-{R}^{2}>0\\ 0,& otherwise\end{array}\right.$$
14
R 2 is the correlation coefficient for a set that contains n substances. R2k is the correlation coefficient for n-1 substances of a set, after removing the k-th substance. Thus, if (R2k - R2) is more than zero, the k-th meaning is an "oppositionist" for the correlation between experimental and predicted values of the set. A small sum of “protests” means a more “intensive” correlation.
Comparison of target functions
Comparison of the determination coefficients on the validation set showed that the best target function is TF3, calculated with Eq. 6.
[Figure 1 around here]
The best model
The best model according to the determination coefficient on the validation set is the following:
DAF = -20.40 (± 1.72) + 0.7427(± 0.0439) * DCW(1,15) (15)
Table 1 lists the statistical characteristics of the model.
Mechanistic interpretation
Table 2 lists promoters for an increase or decrease of DAF. The presence of nitrogen atoms increases DAF; the number of rings from 1 to 4 increases DAF, but four rings with nitrogen and five rings with sulphur reduce it.
Domain of applicability
The domain of applicability for the described model calculated with Eq. 15 defines the so-called statistical defects of SMILES attributes. These defects can be calculated as:
$${d}_{k}=\frac{\left|{P(A}_{k})-{P\text{'}(A}_{k})\right|}{N\left({A}_{k}\right)+N\text{'}\left({A}_{k}\right)}+\frac{\left|{P(A}_{k})-{P\text{'}\text{'}(A}_{k})\right|}{N\left({A}_{k}\right)+N\text{'}\text{'}\left({A}_{k}\right)}+\frac{\left|{P\text{'}(A}_{k})-{P\text{'}\text{'}(A}_{k})\right|}{N\text{'}\left({A}_{k}\right)+N\text{'}\text{'}\left({A}_{k}\right)}$$
16
where P(Ak), P’(Ak) P’’(Ak) are the probability of Ak in the active training set, passive training set, and calibration set, respectively; N(Ak), N’(Ak), and N’’(Ak) are frequencies of Ak in the active training set, passive training set, and calibration set, respectively. The statistical SMILES-defects (Dj) are calculated as:
$${D}_{j}=\sum _{k=1}^{NA}{d}_{k}$$
17
where NA is the number of non-blocked SMILES attributes in the SMILES.
A SMILES falls in the domain of applicability if
$$Dj< 2*\overline{D}$$
In the model calculated with Eq. 15, six SMILES are outside the domain of applicability. However, these do not fall in the validation set.
Comparison with models suggested in the literature
The average statistical characteristics for the validation set for eight random splits studied in Ref. [5] are \(\stackrel{-}{{R}_{v}^{2}}=0.8678, \stackrel{-}{RMSE}=0.391\). In other words, the average statistical characteristics for the model calculated with Eq. 15 are better (Table 1, Fig. 1).
Supplementary materials set out the technical details related to the best model.
The index of ideality of correlation (TF1) and the correlation intensity index (TF2) improve the predictive potential of QSPR for DAF. The simultaneous use of these indices (TF3) is especially effective. The advantage of the TF3 is demonstrated by the considerable absolute average of the determination coefficient on ten random splits and the small dispersion of the value on the ten random splits.
Acknowledgments
The authors are grateful for the contribution of the CONCERT REACH (LIFE17 GIE/IT/000461) for support.
Data availability
The data used in this work and the models developed are freely available Supplementary materials section.
Declaration of competing interests
N/A
Author Contributions: APT was involved in formal analysis, data collection, and curation, investigation, methodology incl. statistics, software, visualization, writing of original draft, and revision. AAT was involved in conceptualization, data curation, formal analysis, investigation, methodology incl. statistics, software, writing of original draft, review, and editing. AR was involved in conceptualization, formal analysis, visualization, writing of original draft, review, and editing. EB was involved in formal analysis, visualization, review, and editing. All authors reviewed the manuscript.
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Table 1
Statistical characteristics of the best model calculated with Eq. 15.
|
Set |
n |
R2 |
CCC |
IIC |
CII |
Q2 |
RMSE |
F |
|
Active training |
18 |
0.7480 |
0.8558 |
0.5504 |
0.8333 |
0.6409 |
2.18 |
47 |
|
Passive training |
18 |
0.7897 |
0.8834 |
0.7324 |
0.9196 |
0.7148 |
1.93 |
60 |
|
Calibration |
18 |
0.9440 |
0.9358 |
0.9707 |
0.9624 |
0.9283 |
1.15 |
270 |
|
Validation |
18 |
0.9468 |
0.9403 |
0.6028 |
0.9628 |
0.9312 |
1.26 |
|
|
All |
72 |
0.8418 |
0.9051 |
0.7716 |
0.9082 |
0.8295 |
1.65 |
|
*) N = number of compounds in a set; R = correlation coefficient; Q = cross-validated R; RMSE = root mean squared error; F = Fischer F-ratio
Table 2
Mechanistic interpretation – promoters of an increase and a decrease of DAF.
|
SMILES attributes SAk |
CWs run 1 |
CWs run 2 |
CWs run 3 |
NA |
NP |
NC |
dk |
|
Promoters of an increase |
|
|
|
|
|
|
|
|
N........... |
1.45058 |
1.24085 |
3.68733 |
18 |
18 |
18 |
0.0000 |
|
\...N....... |
0.86749 |
2.68951 |
0.85243 |
18 |
18 |
18 |
0.0000 |
|
c...(....... |
0.91855 |
0.30987 |
0.02313 |
18 |
18 |
18 |
0.0000 |
|
c........... |
0.59292 |
0.39571 |
0.22744 |
18 |
18 |
18 |
0.0000 |
|
c...1....... |
1.08281 |
0.36335 |
0.35112 |
18 |
18 |
18 |
0.0000 |
|
c...2....... |
1.40532 |
1.21865 |
0.19009 |
18 |
18 |
18 |
0.0000 |
|
c...\....... |
0.86514 |
0.12869 |
0.99491 |
18 |
18 |
18 |
0.0000 |
|
\...(....... |
1.17295 |
1.05545 |
1.01533 |
17 |
17 |
11 |
0.0119 |
|
C........... |
0.74225 |
1.35039 |
1.07951 |
11 |
12 |
4 |
0.0259 |
|
s........... |
0.83436 |
0.85554 |
0.73424 |
11 |
11 |
8 |
0.0088 |
|
c...N....... |
1.10022 |
1.72242 |
1.98520 |
9 |
8 |
7 |
0.0069 |
|
s...c....... |
0.26422 |
0.00880 |
1.19791 |
9 |
7 |
3 |
0.0278 |
|
c...4....... |
0.56773 |
0.49658 |
1.09108 |
7 |
5 |
4 |
0.0152 |
|
-...(....... |
1.13818 |
1.11876 |
1.15964 |
6 |
3 |
8 |
0.0079 |
|
-........... |
1.08951 |
0.63707 |
1.44703 |
6 |
3 |
8 |
0.0079 |
|
Promoters of a decrease |
|
|
|
|
|
|
|
|
=........... |
-0.25405 |
-0.46478 |
-0.60453 |
18 |
18 |
18 |
0.0000 |
|
O...=....... |
-0.14589 |
-0.16774 |
-0.69359 |
18 |
18 |
18 |
0.0000 |
|
S...(....... |
-0.35864 |
-0.78923 |
-0.33920 |
18 |
18 |
18 |
0.0000 |
|
c...O....... |
-0.14306 |
-0.47928 |
-0.31381 |
8 |
9 |
13 |
0.0132 |
|
C...(....... |
-0.09640 |
-0.01961 |
-0.11468 |
4 |
6 |
3 |
0.0079 |
|
n...4....... |
-0.93981 |
-0.16267 |
-0.03242 |
3 |
3 |
3 |
0.0000 |
|
s...5....... |
-0.66599 |
-0.01608 |
-1.16084 |
2 |
3 |
2 |
0.0000 |
*) CWs = correlation weights; NA, NP, and NC are frequencies of a SMILES attribute in the active training set, passive training set; and calibration set, respectively. The dk is the statistical defect of SMILES attribute, calculated via probability in active training set P(SAk), probability in calibration set P’(SAk); frequencies in active training set N(SAk), in calibration set N’(SAk), as the following:
No competing interests reported.