The structures of the examined compounds and their MIC against (i) S. aureus ATCC 25923, and (ii) drug-resistant clinical isolate of S. aureus were taken from previous publications [16, 18, 19, 20, 21, 22]. The molecular structure of the compounds was transferred to the SMILES notation using ACD/ChemSketch software [28]. Due to the structural diversity of examined compounds, the MIC values were recalculated from $\mu$g/mL to mol/L units and expressed as logarithms of reciprocal values. This makes it possible to unify the antibacterial activity of the test substances with respect to the number of molecules (and not the weight). The complete data are represented in Table 1.

The molecular structure can be represented by SMILES and/or a molecular graph (hydrogen suppressed graph). Fig. 2 contains an example of the molecular structure together with the SMILES and the hydrogen suppressed graph for compound 78.

The hybrid optimal descriptors [10] are sensitive to both above-mentioned representations of the molecular structure. Hybrid optimal descriptors are calculated by optimization of the so-called correlation weights of the SMILES attributes together with the correlation weights of the graph invariants. The optimal hybrid descriptor DCW(T,N) is applied for a predictive model of endpoint via the equation:

(1)$$Endpoint=C_0+C_1\times DCW(T,N)$$

(2)$$\mathrm{DCW}(\mathrm{T},\mathrm{N})={\mathrm{DCW}}_{\mathrm{SMILES}}(\mathrm{T},\mathrm{N})+{\mathrm{DCW}}_{\mathrm{graph}}(\mathrm{T},\mathrm{N})$$

where

(3) ${\mathrm{DCW}}_{\mathrm{SMILES}}(\mathrm{T},\mathrm{N})={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{NA}}}\mathrm{CW}\left({\mathrm{S}}_{\mathrm{k}}\right)+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{NA}-1}}\mathrm{CW}\left({\mathrm{SS}}_{\mathrm{k}}\right)$ $+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{NA}-2}}\mathrm{CW}\left({\mathrm{SSS}}_{\mathrm{k}}\right)$

(4) ${\mathrm{DCW}}_{\text{Graph}}(\mathrm{T},\mathrm{N})={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{NG}}}\mathrm{CW}\left({\mathrm{EC1}}_{\mathrm{k}}\right)+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{NG}}}\mathrm{CW}\left({\mathrm{EC2}}_{\mathrm{k}}\right)$ $+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{NG}}}\mathrm{CW}\left({\mathrm{EC3}}_{\mathrm{k}}\right)$

If SMILES = ABCD, the S, SS, and SSS can be represented as

S = (A, B, C, D)

SS = (AB, BC, CD)

SSS = (ABC, BCD)

The EC1, EC2, and EC3 are Morgan extended connectivity of first, second, and third order, respectively. The graph invariants are calculated with the adjacency matrix (Table 2).

The T is an integer to separate SMILES attributes into rare and non-rare. The non-rare SMILES are applied to build up the model. The rare SMILES are not applied to build up the model.

The N is the number of epochs of the optimization of the correlation weights.

The S${}_k$ 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’).

The CW(S${}_k$), CW(SS${}_k$), and CW(SSS${}_k$) are the correlation weights of the above SMILES attributes.

Eqn. 2 needs the numerical data on the above correlation weights. The Monte Carlo optimization is a tool to calculate those correlation weights. Here, two target functions for the Monte Carlo optimization are examined:

(5)$$T{F}_0=r_{AT}+r_{PT}\mathrm{–}\left|r_{AT}\mathrm{–}{r}_{PT}\right|\times 0.1$$

(6)$$TF=T{F}_0+II{C}_C\times 0.5$$

The r${}_{A\mathit{}T}$ and r${}_{P\mathit{}T}$ are the correlation coefficient between the observed and predicted endpoints for the active training set and the passive training set, respectively.

The IIC${}_C$ is the index of ideality of correlation [29, 30], and it is calculated with data on the calibration set as the follows:

(7)$$II{C}_C=r_C\frac{min\mathrm{}{(}^{\mathrm{–}}MA{E}_C{,}^{+}MA{E}_C)}{max\mathrm{}{(}^{\mathrm{–}}MA{E}_C{,}^{+}MA{E}_C)}$$

(8)

(9)$$\max (x,y)=\{\begin{array}{c}x,ifx>y\hfill \\ y,otherwise\hfill \end{array}$$

(10)

(11)$${}^{+}MA{E}_C=\frac{1}{{}^{+}N}\sum \left|{\mathrm{\Delta }}_k\right|{, }^{+}Nisthenumberof{\mathrm{\Delta }}_k\ge 0$$

(12)$${\mathrm{\Delta }}_k=observe{d}_k\mathrm{–}calculate{d}_k$$

The observed and calculated are the corresponding values of the endpoint.

The Monte Carlo optimization that used the IIC${}_C$ is described in the literature [29, 30].

In order to check up the reproducibility of the CORAL [31] models, one should test several splits into the training sub-system (i.e., active training, passive training, and calibration sets) and validation sub-system. The described scheme for three random splits gives the following models:

(a) Split 1:

(13) $\log 1/c(S.a.)=2.363(\pm 0.034)+0.03939(\pm 0.00110)$ $\times DCW(1,15)$

(b) Split 2:

(14) $\log 1/c(S.a.)=2.786(\pm 0.029)+0.02335(\pm 0.00086)$ $\times DCW(1,15)$

(c) Split 3:

(15) $\log 1/c(S.a.)=2.755(\pm 0.026)+0.06817(\pm 0.00178)$ $\times DCW(1,15)$

Table 3 contains the statistical quality of these models. Table 4 (Ref. [32, 33, 34, 35, 36, 37]) contains the statistical criteria of the predictive potential of a model.

For Endpoint 2, the described scheme for three random splits gives the following models:

(a) Split 1:

(16) $\log 1/c(S.a.,\text{isol.})=2.340(\pm 0.019)+0.02649(\pm 0.00037)$ $\times DCW(1,15)$

(b) Split 2:

(17) $\log 1/c(S.a.,\text{isol.})=2.695(\pm 0.017)+0.03511(\pm 0.00060)$ $\times DCW(1,15)$

(c) Split 3:

(18) $\log 1/c(S.a.,\text{isol.})=1.575(\pm 0.056)+0.08254(\pm 0.00227)$ $\times DCW(1,15)$

Table 5 contains the statistical quality of these models.

An example of the technical details for Split 1, i.e., the calculated values for Endpoint 1 (Eqn. 13) and Endpoint 2 (Eqn. 16), and the corresponding correlation weights for the SMILES attributes and graph invariants, is presented in Supplementary Material.

Having numerical data on the correlation weights obtained in several runs of the described Monte Carlo method optimization, one can find molecular features extracted from SMILES or hydrogen suppressed graphs which have solely positive correlation weights. These should be interpreted as promoters of increase for the corresponding endpoint. If a molecular feature has a stable negative correlation weight in several runs of the optimization, it should be interpreted as a promoter of decrease for an endpoint. Table 6 contains a collection of the above promoters for Endpoint 1 and Table 7 contains similar data for Endpoint 2, respectively. One can see (Tables 6,7), that Endpoint 1 and Endpoint 2 have five equivalent promoters (indicated by bold). In other words, these endpoints are far from to be identical ones.

The statistical quality of the models for Endpoint 1 and Endpoint 2 is quite good (Tables 3,5). Reproducibility of the results for both endpoints is observed. However, the predictive potential observed for three random splits is not identical. For both endpoints, the best predictive potential is observed in the case of split 2. The statistical quality of the models for Endpoint 2 is slightly better than that of the models for Endpoint 1. The models suggested here are traditional, that is, multi-targets approach [6, 7, 8], and ADMET [9] are not used here. However, in principle, the approach can be available for the corresponding analyses in the future.