The system of self-consistent models based on quasi-SMILES as a tool to predict the potential of nano-inhibitors of human lung carcinoma cell line A549 for different experimental conditions

Abstract

The different features of the impact of nanoparticles on cells, such as the structure of the core, presence/absence of doping, quality of surface, diameter, and dose, were used to define quasi-SMILES, a line of symbols encoded the above physicochemical features of the impact of nanoparticles. The correlation weight for each code in the quasi-SMILES has been calculated by the Monte Carlo method. The descriptor, which is the sum of the correlation weights, is the basis for a one-variable model of the biological activity of nano-inhibitors of human lung carcinoma cell line A549. The system of models obtained by the above scheme was checked on the self-consistence, i.e., reproducing the statistical quality of these models observed for different distributions of available nanomaterials into the training and validation sets. The computational experiments confirm the excellent potential of the approach as a tool to predict the impact of nanomaterials under different experimental conditions. In conclusion, our model is a self-consistent model system that provides a user to assess the reliability of the statistical quality of the used approach.

Keywords:

• Computational nano-toxicology
• human lung carcinoma cell line A549
• system of self-consistent models
• quasi-SMILES
• CORAL software

Introduction

The application of nanoparticles in medicine (Singh et al. 2020), the ecological impact of nanomaterials (Kleandrova et al. 2014), and nano-toxicology (Alfaro-Moreno et al. 2013) are new and intensively developing fields of knowledge in natural sciences. Expanding the range of nanomaterials leads to very effective developments in the chemical industry, electronics, and medicine. Still, it also leads to an increase in ecological and human health safety problems (Toropova and Toropov 2022). Nanomaterials are expected to be applied in a large number of variations in properties such as size, shape, coating, and chemical composition, and properties without analogies at present (Zielińska et al. 2020, Lebre et al. 2022). It is possible to detect a group of new special endpoints related to nanomaterials’ genetic and ecological impact (e.g., carcinogenicity). The complexity of nanomaterials leads to problems in solving the task of assessing their safety and potential risks. The levels of exposure to nanomaterials under different conditions caused the necessity to develop tools that allow reliable indicators of dangerous nanomaterials (Panneerselvam and Choi 2014, Lebre et al. 2022).

Like traditional substances, nanomaterials desire computational interpretation of their physicochemical and biochemical behavior. However, if quantitative structure–property/activity relationships (QSPRs/QSARs) for traditional substances are a fundamental research tool, nano-QSPR/QSAR has not yet become such a tool. The classic QSPR/QSAR analysis is based on different approaches representing the molecular structure and, as a rule, aims to select substances that are most suitable candidates for defined aims. Statistical standardization of the analysis of the physicochemical and biochemical behavior of nanomaterials, as well as for traditional substances, can be implemented through compliance with the so-called five principles of the OECD (Organization for Economic Cooperation and Development): (1) an endpoint must be defined; (2) the exact algorithm of calculation of the model is formulated; (3) the area of applicability of the model is clearly indicated; (4) statistical measures of model reliability are indicated; (5) finally, a mechanistic interpretation of the model is quite desirable, if possible (OECD 2014, 2020; Toropova and Toropov 2022). The molecular graph (Mercader et al. 2000) data on vectors of physicochemical parameters (Polyakova et al. 2006) were the most popular basis for QSPR/QSAR works from 1980x to 2000x. Later, the simplified molecular input-line entry system (SMILES) (Weininger 1988) became a novel representation of the molecular structure for the QSPR/QSAR (Toropov et al. 2005).

The impact of nanomaterials on different objects (organs, organisms, and ecosystems) is often defined by somewhat experimental conditions, not by molecular structure. Under such circumstances, the quasi-SMILES (Toropov and Toropova 2015) become the more informative basis for building up models of physicochemical and biochemical behavior of nanomaterials, because those reflect the impact of experimental conditions (Trinh et al. 2018; Toropov and Toropova 2019, 2021a; Ahmadi et al. 2021; Toropova et al. 2021; Bunmahotama et al. 2022) as well as for traditional substances (Toropov et al. 2016). The present study aimed to estimate nano-QSAR models for viability cell line A549 under inhibition by various nanoparticles and under different physicochemical conditions.

Methods

Data

Experimental data (raw) on inhibitors’ impact on A549 cells by different nanoparticles under different experimental conditions (structure of the core, doping, surface, diameter, and dose) are taken from the literature (Shin et al. 2021). A collection of the different practical situations of the influence of different nanoparticles under the above experimental conditions contains 377 samples.

These samples are randomly split into active training (≈25%), passive training (≈25%), calibration (≈25%), and validation sets (≈25%). Table 1 confirms that the five random splits examined here are far from identical. Each of the above subsets has its own task (Toropova and Toropov 2022). The task for the active training set is to compute correlation weights that provide a correlation between the experimental and calculated endpoint for the active training set. The task of the passive training set is to check whether the correlation obtained for the active training set is confirmed for similar compounds distributed in the passive training set. The calibration set is used to determine the start of overtraining. The final assessment of the predictive potential of the model is the task for the external validation set (invisible during model building).

Table 1. Percentage of identical distribution of samples into the active/passive training sets, calibration set, and validation set.

	Split 1	Split 2	Split 3	Split 4	Split 5
Split 1	0	40.6	46.0	40.2	37.9
Split 2	43.5	0	36.8	42.8	33.2
Split 3	43.8	43.5	0	39.6	40.4
Split 4	51.6	40.8	36.8	0	41.1
Split 5	40.2	41.1	39.2	38.5	0

Matrix Element[i,j], i>j the identity for the active training sets. Matrix Element[i,j], i<j the identity for the validation sets.

$\mathrm{Identity} \left(\mathrm{\%}\right)=\frac{N_{ij}}{0.5\times (N_i+N_j)}\times 100.$

N_ij is the number of quasi-SMILES that are distributed into the same set for both ith. split and jth split (set = active training set or passive training set or calibration set or validation set). N_i is the total number of substances that are distributed into the set for ith split. N_j is the total number of substances that are distributed into the set for jth split.

The cell viability serves as a measure of the biological activity of the above samples. The endpoint is expressed in μg/ml.

Quasi-SMILES building up

Figure 1 contains the general scheme of building up of quasi-SMILES. The traditional SMILES contain information on the 2D structures of molecules representing a nanoparticle’s core. The quasi-SMILES are lines of symbols that encode data on molecular structure (if it is expedient) and experimental conditions (if reasonable). Thus, the quasi-SMILES is a basis for building up a model as a mathematical function of the molecular structure of the core of the nanoparticle and physicochemical conditions of acting the nanoparticle: (1) $$\mathrm{Endpoint}=F (\text{molecular structure},\text{ experimental conditions})$$(1)

Figure 1. The general scheme of building up quasi-SMILES.

The general scheme of building up models

Models of cell viability of nanoparticles under different experimental conditions are calculated as follows: (2) $${\text{Cell viability}[\mathrm{\mu }\mathrm{g}/\mathrm{ml}]=C}_0+C_1\times \mathit{DCW}(T,N)$$(2)

The C₀ and C₁ are regression coefficients. The DCW(T,N) is the optimal descriptor calculated with quasi-SMILES. The descriptor is calculated as follows: (3) $$\mathit{DCW}\left(T,N\right)=\sum CW\left({\mathit{APP}}_k\right)+\sum CW\left(S_k\right)+\sum {CW(SS}_k)+\sum CW\left({qS}_k\right)$$(3)

The APP_k is the vector of atoms pair’s proportions (Toropova, Toropov, and Benfenati 2021) related to oxygen (‘O’) and double bonds (‘═’) proportions. There are six APP_k codes. These are the following: (O.═).1.1 (this code means that one oxygen and one double bond are present in a sample); (O.═).2.2; (O.═).2.3 (this code means that two oxygens and three double bonds are present in a sample); (O.═).3.2; (O.═).4.3; and (O.═).6.3.

The S_k is a so-called SMILES atom. The SMILES atom is one symbol (e.g., ‘C’, ‘O’, ‘N’, etc.) or a group of symbols which cannot examine separately (e.g., ‘Cl’, ‘Br’, 12%, etc.) (Weininger 1988). The SS_k are combinations of two SMILES atoms, respectively. The qS_k is an atom of quasi-SMILES (Figure 1). The CWs for the listed attributes of SMILES and quasi-SMILES are calculated by the scheme described in the literature (Toropova, Toropov, and Benfenati 2021).

The threshold value T is used to classify the correlation weights. If the frequency of the corresponding attribute in the active training set is less than T, then this attribute is considered ‘rare’ and excluded from the model building process. In other words, correlation weights for rare features are defined as zero (blocked attribute). N is the number of Monte Carlo optimization epochs. Here, T= 1 and N= 15.

The Monte Carlo optimization aims to provide the correlation weights that produce the maximal value for the target function suggested in the literature, where the so-called index of ideality of correlation is applied (Toropova, Toropov, and Benfenati 2021).

Two target functions were compared as the basis for the Monte Carlo optimization. They are as follows: (4) $${TF}_1=r_{AT}+r_{PT}-\left|r_{AT}-r_{PT}\right|\times 0.1$$(4) (5) $${TF}_2={TF}_1+{\mathit{IIC}}_{ }\times 0.5$$(5)

The $r_{AT}$ and $r_{PT}$ are correlation coefficients between the observed and predicted endpoint for the active and passive training sets, respectively. The IIC is the index of ideality of correlation (Toropova and Toropov 2017). The calculation of the IIC is as follows: (6) $${\mathit{IIC}}_C=r_C\frac{\min (-{\mathit{MAE}}_C,+{\mathit{MAE}}_C)}{\max (-{\mathit{MAE}}_C,+{\mathit{MAE}}_C)}$$(6) (7) $$\min \hspace{0.17em}\left(x,y\right)=\{\begin{array}{c}x,\mathit{\text{ if x}}<y\\ y,\mathit{otherwise}\end{array}$$(7) (8) $$\max \hspace{0.17em}\left(x,y\right)=\{\begin{array}{c}x,\mathit{\text{ if x}}>y\\ y,\mathit{otherwise}\end{array}$$(8) (9) $$-{\mathit{MAE}}_C=\frac{1}{{}_{ }{}^{-}N}\sum \left|{\Delta }_k\right|,-N\text{ is the number of}{\Delta }_k<0$$(9) (10) $$+{\mathit{MAE}}_C=\frac{1}{{}_{ }{}^{+}N}\sum \left|{\Delta }_k\right|,+N\text{ is the number of}{\Delta }_k\ge 0$$(10) (11) $${\Delta }_k={\mathrm{observed}}_k-{\mathrm{calculated}}_k$$(11)

The observed_k and calculated_k are the corresponding values of the endpoint.

Figure 2 shows an example of the model for split 1. It should be noted that despite the statistical quality of the model for the active and passive training sets being low, these sets contain two latent correlations indicated by red and green. Apparently, this is the effect of exposure to IIC. Analogical pairs of correlations were observed in computer experiments described in the literature (Toropov et al. 2022). Figure 2 indicates that latent correlations are statistically significant.

Figure 2. The graphical representation of the model was obtained for split #1 by the CORAL program interface. Red and green colors show the statistical characteristics of hidden correlations for active and passive training sets.

The system of self-consistent models

Each ith model has ith validation set. As shown in Table 1, the results of the validation sets are not identical. It is critical to determine if an ‘arbitrary model’ can be used for an ‘arbitrary validation set’ (otherwise, the model is only suitable for the validation set used to build that model). If the answer is yes, these different models should be considered as ‘self-consistent’ ones (Toropova et al. 2022).

A measure of self-consistency is the mean value and variance of the correlation coefficient observed on different validation sets. The matrix is a diagram of the corresponding computational experiments: (12) $$\left[\begin{array}{ccc}(M_1:V_1\to {Rv}_{11}^2)& \cdots & {(M}_5:V_1\to {Rv}_{51}^2)\\ ⋮& & ⋮\\ {(M}_1:V_5\to {Rv}_{15}^2)& \cdots & (M_5:V_5\to {Rv}_{55}^2)\end{array}\right]$$(12) the $M_i$ is an ith model; the $V_j$ is the list of nanoparticles applied as the validation set in the case of jth split; the ${{Rv}_{ij}^2}_{\mathit{}}$ is the correlation coefficient observed for the jth validation set if applied ith model.

Results

Similar models for the inhibitory activity of nanoparticles under different experimental conditions for human lung carcinoma cell line A549 are unavailable. The original work developed local models to predict 32 nanoparticles without directly indicating lists for the training and test sets.

Table 2 contains the statistical characteristics of the models calculated for five random splits.

Table 2. The statistical quality of models for splits #1–#5 observed in the cases of the Monte Carlo optimization with different target functions calculated with Equation (4)(4) $${TF}_1=r_{AT}+r_{PT}-\left|r_{AT}-r_{PT}\right|\times 0.1$$(4) and Equation (5)(5) $${TF}_2={TF}_1+{\mathit{IIC}}_{ }\times 0.5$$(5) .

Split	Target function	Set	n^a	R^2	CCC	IIC	Q^2	RMSE	MAE	F
1	TF1	Active training	92	0.5989	0.7491	0.6221	0.5772	24.8	20.4	134
		Passive training	97	0.5142	0.7043	0.6703	0.4838	25.9	21.2	101
		Calibration	92	0.1720	0.2262	0.2789	0.1248	38.8	32.5	19
		Validation	96	0.2702				33.5	28.9
2		Active training	95	0.5680	0.7245	0.5481	0.5412	21.0	16.6	122
		Passive training	91	0.6022	0.7355	0.4788	0.5820	29.0	22.6	135
		Calibration	94	0.1552	0.2133	0.2416	0.1100	31.6	25.6	17
		Validation	97	0.1549				32.6	27.0
3		Active training	95	0.5889	0.7413	0.7514	0.5689	23.7	20.2	133
		Passive training	94	0.5888	0.7124	0.5347	0.5670	24.5	20.9	132
		Calibration	92	0.0497	0.1699	0.2057	0	28.2	23.7	5
		Validation	96	0.2437				27.8	24.2
4		Active training	92	0.5025	0.6688	0.6787	0.4757	24.2	20.3	91
		Passive training	98	0.6097	0.6618	0.5156	0.5886	27.4	24.1	150
		Calibration	93	0.1022	0.1854	0.2251	0.0411	25.7	22.0	10
		Validation	94	0.3487				22.5	18.9
5		Active training	98	0.5341	0.6963	0.5040	0.5105	24.1	20.4	110
		Passive training	95	0.5645	0.6597	0.7030	0.5419	27.3	24.5	121
		Calibration	91	0.2440	0.4712	0.4139	0.1959	20.6	16.9	29
		Validation	93	0.3430				21.1	17.3
1	TF2	Active training	92	0.3910	0.5622	0.5253	0.3620	30.5	27.7	58
		Passive training	97	0.3983	0.5606	0.5660	0.3642	28.6	25.8	63
		Calibration	92	0.6400	0.7844	0.7999	0.6062	11.0	9.07	160
		Validation	96	0.6611				11.6	9.23
2		Active training	95	0.3202	0.4851	0.5093	0.2856	26.3	23.1	44
		Passive training	91	0.5153	0.5334	0.4602	0.4932	33.4	29.3	95
		Calibration	94	0.6834	0.8006	0.8267	0.6624	8.54	6.78	199
		Validation	97	0.5728				9.39	7.86
3		Active training	95	0.3969	0.5683	0.5436	0.3695	28.7	25.0	61
		Passive training	94	0.4179	0.5320	0.4621	0.3851	29.3	25.7	66
		Calibration	92	0.5264	0.6868	0.7255	0.4814	12.8	10.2	100
		Validation	96	0.6796				11.6	9.18
4		Active training	92	0.3637	0.5334	0.5066	0.3300	27.4	24.0	51
		Passive training	98	0.4482	0.5157	0.4907	0.4209	31.9	28.6	78
		Calibration	93	0.5952	0.6671	0.7715	0.5618	11.1	9.20	134
		Validation	94	0.6603				10.3	8.59
5		Active training	98	0.3885	0.5596	0.5292	0.3595	27.6	23.6	61
		Passive training	95	0.4232	0.5440	0.5507	0.3941	30.4	26.0	68
		Calibration	91	0.5386	0.7315	0.7339	0.5017	12.5	10.1	104
		Validation	93	0.5134				12.4	9.89

a n: the number of quasi-SMILES in a set; R²: the determination coefficient; CCC: concordance correlation coefficient; IIC: index of ideality of correlation; Q²: cross-validated R²; RMSE: root mean squared error; MAE: mean absolute error; F: Fisher’s F-ratio.

These results indicate that the models developed are not perfect because the statistical quality of these models is not high. Moreover, the allocation to the training set (here, it is structured into three sets, namely, active learning, passive learning, and calibration sets) has a noticeable effect on the predictive potential of the models. However, it is important that the correlation ideality index has a clear impact on the predictive potential. The use of the mentioned index in Monte Carlo optimization leads to a deterioration in the statistics for the training samples (excluding the calibration set) but to a clear improvement in the statistics for the external validation set.

This paradoxical situation is accompanied by dividing point arrays on the active and passive training samples into pairs of fairly significant separate correlations (Figure 2). Thus, the resulting models can be evaluated as semi-quantitative. Their important quality is reproducibility. It is possible that with the involvement of additional initial information, the quality of the models will be improved. Finally, it is important that for all the considered models for five random partitions, the coefficient of determination for all 377 quasi-SMILES when using TF₂ is greater than when using the objective function TF₁. Thus, the expediency of using the index of ideality of correlation is once again confirmed.

For biochemical phenomena, high values of correlation coefficients are rare and may seem ‘artificial’ if considering the wide diversity of factors affecting them. In this work, correlations for external datasets are significant. These values are enough to state quite good replicability on five random splits. Perhaps by taking into account some additional experimental data, these models will be improved. The ability to use quasi-SMILES as a language of communication between experimenters and developers of models is another important aspect of the application of Quasi-SMILES (Toropova et al. 2022; Toropova, Toropov, and Fjodorova 2022; Toropov, Kjeldsen, and Toropova 2022).

Applicability domain

Applicability domain for models calculated by the CORAL software was defined according to SMILES and quasi-SMILES attributes’ statistical defects (Toropova, Toropov, and Benfenati 2021; Toropova et al. 2022), i.e., according to their frequencies in the active-, passive training sets, as well as their frequencies in the calibration set (validation set is external one).

In other words, the applicability domain for the described models was defined via the so-called statistical defects of quasi-SMILES attributes calculated as: (13) $$d_k=\frac{\left|{P(A}_k)-{P\prime (A}_k)\right|}{N\left(A_k\right)+N\prime \left(A_k\right)}+\frac{\left|{P(A}_k)-{P″(A}_k)\right|}{N\left(A_k\right)+N″\left(A_k\right)}+\frac{\left|{P\prime (A}_k)-{P″(A}_k)\right|}{N\prime \left(A_k\right)+N″\left(A_k\right)}$$(13) where P(A_k), P′(A_k) P″(A_k) are the probability of A_k in the active training set, passive training set, and calibration set, respectively; N(A_k), N′(A_k), and N″(A_k) are frequencies of A_k in the active training set, passive training set, and calibration set, respectively. The statistical quasi-SMILES-defects (D_j) are defined as: (14) $$D_j=\sum _{k=1}^{NA}{d}_k$$(14) where NA is the number of non-blocked quasi-SMILES attributes in the quasi-SMILES.

A quasi-SMILES falls in the applicability domain, if (15) $$Dj< 2\mathrm{*}\overline{D}$$(15)

The $\overline{D}$ is the average value of the statistical defect on active and passive training sets.

Hence, the definition of the applicability domain is advisory in nature. However, very large values of the statistical defect are an obvious reason to exclude quasi-SMILES from the applicability domain of the model. The percentage of ‘unreliable’ quasi-SMILES depends on the distributions of the available data to the active training, passive training, calibration, and validation sets. The specified percentage (according to inequality 9) was 17, 23, 23, 17, and 25 for splits #1, #2, #3, #4, and #5, respectively.

Mechanistic interpretation

If several trials of Monte Carlo optimization are carried out, different sets of correlation weights will be obtained for the same attributes of quasi-SMILES. Table 3 contains the results of such calculations. Those attributes that have solely positive correlation weights should be assessed as contributors to the increase in the endpoint in question. Those attributes of quasi-SMILES that have solely negative weights should be considered as contributors to reducing the simulated endpoint. It should be noted that attribute frequencies must also be taken into account. Those attributes, which are rare in the active and passive training sets, are unlikely to be the reliable basis for any statistical hypotheses. Thus, the absence of doping [NoDop], the presence of two oxygen atoms and two double bonds in the nanoparticle’s core that is represented by code ‘(O.═).2.2.’ as well as the presence of silver [Ag] should lead to an increase in inhibitory activity. In contrast, large doses indicated as [do100] and [do50] should reduce inhibitory activity.

Table 3. Fragments of quasi-SMILES are promoters of increase/decrease inhibition of human lung carcinoma cell line A549.

APP,Sk,SSk,qSk	CWs Run 1	CWs Run 2	CWs Run 3	Na	Np	Nc	Statistical defect, dk
[NoDop]…..	0.24666	1.04451	1.19031	48	63	86	0.0031
(O..═)..2.2.	0.90088	1.54215	1.46468	16	15	31	0.0035
[Ag]……..	2.24721	0.85776	1.92846	15	11	15	0.0000
[do6,25]….	2.36345	1.06771	1.56749	15	12	9	0.0027
[do0,01]….	2.71251	1.59610	2.12650	12	9	10	0.0010
[do12,50]…	2.08471	1.60590	2.11192	11	11	16	0.0020
[do25,00]…	2.85563	1.65641	2.34717	11	8	15	0.0017
[SOCIT]…..	2.39803	1.09635	2.37297	9	13	18	0.0036
[PVP]…….	0.28446	0.58024	0.68047	8	10	6	0.0016
[Ce]……..	1.72872	1.00184	1.31621	5	2	2	0.0047
[Fe]……..	0.82665	0.49958	0.58206	5	14	14	0.0051
(………..	0.33099	0.07855	0.78283	3	1	5	0.0027
C…(…….	0.93386	0.93735	0.95397	3	1	5	0.0027
[Au(25%)]…	0.58743	1.55060	1.73304	3	0	1	0.0054
[Citrate]…	1.24923	0.53949	2.00244	3	0	2	0.0022
[do100,00]..	–0.12733	–0.56522	–0.57266	21	19	3	0.0082
[do50,00]…	–0.47533	–0.24178	–0.97419	16	17	11	0.0020
[Cl(3%)]….	–0.81813	–0.21497	–0.07199	4	0	1	0.0065
[Co(2%)]….	–0.22639	–0.16728	–0.25761	4	2	0	1.0000
C…C…….	–0.36947	–0.69105	–0.42154	3	1	5	0.0027
[COOH]……	–0.35344	–0.35271	–0.52850	3	1	7	0.0043
[di149,00]..	–1.20930	–0.13487	–0.54397	3	3	0	1.0000
[di164,70]..	–0.20162	–0.41797	–0.21815	3	3	0	1.0000
[bPEI]……	–1.26530	–0.44992	–0.96465	2	3	1	0.0036
[di1968,00].	–1.46506	–0.90406	–0.52725	2	4	0	1.0000
[di28,55]…	–1.24826	–0.56019	–1.15220	2	3	1	0.0036
(O..═)..3.2.	–1.26318	–0.81951	–0.22331	1	4	2	0.0036
[Cr]……..	–1.14221	–0.79011	–1.33245	1	1	1	0.0000
[di1159,00].	–0.76851	–0.46016	–1.29132	1	1	1	0.0000

Na, Np, and Nc are the numbers of quasi-SMILES that contain the code in the active training, passive training sets, and calibration set, respectively.

The predictive potential

The self-consistency of models (Toropov and Toropova 2021b, Toropova and Toropov 2021) obtained for different splits into the training (visible) and validation (invisible) sets confirms the good quality (predictive potential) suggested models. The same calculations without APP and IIC give poor values for the determination coefficient for diverse validation sets. One can see that the current version (Table 4) of the design of models provides the above value of about 0.6 or even more significant. The determination coefficient values represented in Table 4 is accompanied by the number of absent compounds in corresponding active training, passive training, and calibration sets.

Table 4. The statistical characteristics of the models for the lists of quasi-SMILES that were unknown when developing the corresponding model (i.e., quasi-SMILES which are absent in the active training, passive training, and calibration sets used to develop corresponding models).

	Split 1	Split 2	Split 3	Split 4	Split 5
Model1		n = 42, $R_{\mathit{VLD}}^2=0.69, \mathit{MAE}=6.6$^a	n = 42, $R_{\mathit{VLD}}^2=0.77, \mathit{MAE}=7.2$	n = 49, $R_{\mathit{VLD}}^2=0.67, \mathit{MAE}=7.9$	n = 38, $R_{\mathit{VLD}}^2=0.62, \mathit{MAE}=7.6$
Model2	n = 42, $R_{\mathit{VLD}}^2=0.72, \mathit{MAE}=9.0$		n = 42, $R_{\mathit{VLD}}^2=0.63, \mathit{MAE}=8.0$	n = 39, $R_{\mathit{VLD}}^2=0.46, \mathit{MAE}=7.5$	n = 39, $R_{\mathit{VLD}}^2=0.65, \mathit{MAE}=8.7$
Model3	n = 42, $R_{\mathit{VLD}}^2=0.81,$ MAE = 8.4	n = 42, $R_{\mathit{VLD}}^2=0.72, \mathit{MAE}=7.8$		n = 35, $R_{\mathit{VLD}}^2=0.69, \mathit{MAE}=8.6$	n = 37, $R_{\mathit{VLD}}^2=0.61, \mathit{MAE}=8.3$
Model4	n = 49, $R_{\mathit{VLD}}^2=0.67, \mathit{MAE}=9.0$	n = 39, $R_{\mathit{VLD}}^2=0.54, \mathit{MAE}=7.1$	n = 35, $R_{\mathit{VLD}}^2=0.65, \mathit{MAE}=7.8$		n = 36, $R_{\mathit{VLD}}^2=0.69, \mathit{MAE}=8.1$
Model5	n = 38, $R_{\mathit{VLD}}^2=0.57, \mathit{MAE}=8.3$	n = 39, $R_{\mathit{VLD}}^2=0.46, \mathit{MAE}=7.2$	n = 37, $R_{\mathit{VLD}}^2=0.54, \mathit{MAE}=7.7$	n = 36, $R_{\mathit{VLD}}^2=0.72, \mathit{MAE}=8.8$

a n: the number of quasi-SMILES in the validation set of defined splits that are not involved in the active training, passive training, and calibration sets applied to build the corresponding model. R²_VLD: determination coefficient for quasi-SMILES selected by the above limitations. MAE: mean absolute error for indicated lists of quasi-SMILES.

Supplementary materials section contains details on the model for split 1, i.e., the experimental and calculated cell viability (μg/ml), together with lists of quasi-SMILES distributed in the active training, passive training, calibration, and validation sets. In addition, the list of abbreviations applied to construct quasi-SMILES codes is represented in the Supplementary material section.

Conclusions

The model of the biological activity of nano-inhibitors of human lung carcinoma cell line A549, calculated with data on the structure of the nanoparticle core and data on physicochemical conditions of the experiment, can be a reliable basis for building up a prediction of the endpoint. The use of the index of ideality of correlation in the Monte Carlo optimization process significantly improves the statistical quality of the model for the calibration set and the validation set since the use of this index blocks overtraining, i.e., it avoids the situation when the statistical quality on the training set is noticeably higher than on the external validation set. The self-consistent model system provides a user to assess the reliability of the statistical quality of the used approach for a group of different random splits into the training and test sets.

Author contributions

APT was involved in formal analysis, data collection, curation, investigation, methodology including statistics, software, visualization, writing of original draft, and revision. AAT was involved in conceptualization, data curation, formal analysis, investigation, methodology including statistics, software, writing of original draft, review, and editing. JM was involved in conceptualization, formal analysis, visualization, writing of original draft, review, and editing. EAM was involved in formal analysis, visualization, review, and editing. All authors revised it critically and finally approved the version to be submitted.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

The data used in this work and developed models are freely available Supplementary materials section.

Additional information

Funding

AAT and APT are grateful to the project LIFE-CONCERT (LIFE17 GIE/IT/000461) for their support. EAM was supported by European Union’s H2020 project Sinfonia (N.857253). JM was supported by SbDToolBox, NORTE-01-0145-FEDER-000047, supported by Norte Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund.