Local potential energy density-supramolecular energy (LPED-SME) machine learning prediction – a web application to obtain the local SME from simple inputs

doi:10.21203/rs.3.rs-4945250/v1

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Local potential energy density-supramolecular energy (LPED-SME) machine learning prediction – a web application to obtain the local SME from simple inputs

https://doi.org/10.21203/rs.3.rs-4945250/v1

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We developed a Flask web application that uses supervised machine learning (ML) to predict the local potential energy density (LPED) based on intermolecular and intramolecular interactions. The predictions are made from simple inputs, specifically the atomic charges of interacting atoms (using MK, ChelpG, or RESP schemes) and the interatomic distances between them. This application streamlines the process by avoiding the more complex calculations required by QTAIM topology. We optimized the size of our dataset to 53 samples, being a simple dataset with only three numerical features and no categorical features. We tested five different ML models and found that Linear Regression performed the best, achieving an R² score of 0.88, a mean absolute error (MAE) of 0.72 kcal/mol·Bohr³, a mean squared error (MSE) of 0.82 kcal²/mol²·Bohr⁶, and a root mean squared error (RMSE) of 0.91 kcal/mol·Bohr³. To ensure the reliability of our model, we conducted a secondary validation using a different set of input data with known LPED values. The predicted values closely matched the actual values, and the metrics from this secondary validation were similar to those from the primary testing. With this double validation, our web application is a reliable tool for obtaining LPED and local supramolecular energy (SME) from straightforward inputs. The major physical insight is the capability of the machine learning model to obtain a topologically derived information such as LPED using non-topological data.

QTAIM

Local potential energy density

LPED

supramolecular binding energy

Machine Learning

Linear Regression Model

The electrostatic force (Eq. 1) can be used to explain several physical-chemical properties such as solubility, boiling points, resonance, and chemical bond strengths, which are associated with the intermolecular interactions or the electronic nature of the studied system[1]. Accordingly, the potential energy from Coloumb’s law (Eq. 2) is the most important component for analyzing the molecular energy of an optimized system according to the virial theorem[2], where the potential energy (V) is twice the kinetic energy of a system.

$$\:{\overrightarrow{F}}_{12}=K\frac{{q}_{1}{q}_{2}}{{r}_{12}^{3}}{\overrightarrow{r}}_{12}\:\:\:\left(1\right)$$

$$\:{V}_{C}=K\frac{{q}_{1}{q}_{2}}{{r}_{12}}\:\:\:\left(2\right)$$

Where K is the Coulomb’s constant (9 x 10⁹ N m² C^− 2).

Bader[3] applied the virial theorem to its quantum theory of atoms in molecules (QTAIM) whose applicability was later improved.[4] Recently, we developed the local potential energy density, LPED,[5, 6] which is based on Coulomb’s law for two charged particles using topological data from QTAIM. The LPED provides the energy stored in a particular intra/intermolecular interaction per unit volume (Eq. 3).

$$\:{LPED}_{interaction}=\frac{1}{2}\left(\frac{{\rho\:}_{bcp}{q}_{N\left(I\right)}}{{r}_{I-bcp}}+\frac{{\rho\:}_{bcp}{q}_{N\left(II\right)}}{{r}_{II-bcp}}\right)=\:\frac{1}{2}\left(\frac{{q}_{bcp}{Z}_{eff\left(I\right)}}{{V}_{bcp}{r}_{I-bcp}}+\frac{{q}_{bcp}{Z}_{eff\left(II\right)}}{{{V}_{bcp}r}_{II-bcp}}\right)\:\left(3\right)$$

Where I and II are the interacting atomic basins of the intermolecular or intramolecular interaction; ρ_bcp is the charge density of the bond critical point; r_I−bcp and r_II−bcp are the distance of the bond path from the bond critical point to the nuclear attractor of the interacting atomic basins I and II, in Bohr unit (where Bohr radius corresponds to 0.529177 Angstroms), in the corresponding bond path, and Z_eff(I/II) rewritten to Z_eff(i), in a general form, is the QTAIM effective atomic number of the ith atomic basin (Eq. 4).

$$\:{Z}_{eff\left(i\right)}={N}_{i}-{LI}_{i}=\left({Z}_{i}-{q}_{i}\right)-{LI}_{i}\left(4\right)$$

Where q_i is the QTAIM atomic charge of the ith atomic basin, N_i is the total number of electrons of the ith atomic basin, and LI_i is the localization index of the ith atomic basin.

The values of the LPED of several intermolecular interactions from more than 20 investigated molecular complexes provide an excellent correlation with their corresponding supramolecular energy (the energy of interaction between two molecules), SME, with 94% linearity.[7] Then, after geometry optimization and QTAIM calculations of a molecular complex, it is possible to obtain the corresponding local and total supramolecular energy of a system with multiple intermolecular interactions. However, obtaining the LPED of local intermolecular interactions can be difficult for non-familiar users with QTAIM. Aiming to facilitate the obtention of LPED and, consequently, the SME, we applied a machine learning model to provide LPED from simpler inputs: interatomic distance and atomic charges of the interacting atoms from the optimized geometry of a molecular complex.

In this work, we created a Flask framework[8] application from the Python language and Scikit-Learn library.[9] We used a linear regression model to perform a machine learning train-test predictive analysis. Part of the dataset came from the complexes we have analyzed in our previous works,[5–7] and another part was added to this work, totaling 53 samples. The coefficient of determination (R²) in the test-predictive analysis (primary testing) was 0.88, and the root mean squared error (RMSE) was 0.91 kcal mol^− 1 Bohr^− 3.

Our web application is hosted in PythonAnywhere repository.[10] The physical insight of this work is the use of geometric parameters (interatomic distance) and the electrostatic potential atomic charges to obtain, via ML modeling, the values of LPED which are derived from topological data such as charge density of bond critical points, bond paths, and effective atomic numbers from localization index and QTAIM atomic charge.

RATIONALE ON THE TESTED MACHINE LEARNING MODELS

We tested five different machine-learning models to find the one with the best metrics for our application. They were linear regression, decision tree regressor, random forest regressor, gradient boosting regressor, and XGB regressor.

Linear regression is a supervised machine learning algorithm that models the relationship between a dependent variable and one or more independent variables as a linear equation.[11] It aims to find the best-fitting line (or hyperplane for multiple variables) that minimizes the difference between the predicted values and the actual values. The model coefficients represent the weight of each independent variable in influencing the dependent variable. The algorithm learns these coefficients by minimizing a cost function, typically the mean squared error (MSE).

Decision tree regression is a supervised machine learning algorithm that constructs a tree-like structure to predict continuous target values[12]. It recursively splits the data into subsets based on the values of independent variables, creating branches and nodes. Each node represents a decision rule based on a specific feature, and each leaf node holds a predicted value for the target variable. The algorithm selects the best split at each node by minimizing a cost function, such as mean squared error.

Random forest regression[13] is an ensemble learning method that combines multiple decision trees to predict continuous target values. It operates by building a multitude of decision trees on different random subsets of the training data and features. Each tree predicts a value, and the final prediction is the average of all individual tree predictions. This averaging process reduces variance and improves the model's generalization ability by mitigating overfitting.

Gradient Boosting Regressor[14] is an ensemble machine learning model used for regression tasks that combines the predictions of several weak learners, typically decision trees, to produce a strong overall model. It works by sequentially adding trees, where each new tree aims to correct the errors made by the previous ones through gradient descent optimization. By adjusting the weights of the data points based on the residual errors, Gradient Boosting minimizes the loss function, often resulting in highly accurate predictions.

Both Gradient Boosting Regressor and Random Forest are ensemble learning models derived from the Decision Tree model. Random Forest creates an ensemble by constructing multiple decision trees during training and averaging their predictions to improve accuracy and prevent overfitting. In contrast, Gradient Boosting Regressor builds an ensemble by sequentially adding decision trees, where each new tree attempts to correct the errors of the preceding ones through gradient descent optimization.

All the above-mentioned models are implemented in the Scikit-Learn Python library[15] which is used in this work.

XGBoost (eXtreme Gradient Boosting) Regressor is an improved version of Gradient Boosting. XGBoost[16] builds an ensemble of trees sequentially, where each tree attempts to correct the errors of the preceding ones by minimizing a specified loss function. It boasts features like regularization to prevent overfitting, tree pruning, and handling missing values.

COMPUTATIONAL DETAILS FOR BUILDING THE DATASET

The dataset for the machine learning modeling was based on the optimized geometries of molecular complexes studied in our previous works related to LPED in addition to other complexes that we analyzed in this work aiming to obtain the optimized size of the dataset and the highest precision of the ML modeling. Other complexes were analyzed for secondary testing. Then, the additional complexes for the dataset are 1 to 15 and the complexes for secondary testing are 16 to 21. The geometry optimization the complexes 1 to 21 was performed using ωB97X-D/6-311 + + G(2d,2p) level of theory [17, 18] in Gaussian 09 software.[19] Frequency analysis was done subsequently to confirm that these structures were, in fact, minimum at the potential energy surface. From the optimized geometries, we have obtained their corresponding wave function (wfn) files for the QTAIM calculations of their LPED using AIM2000 software.[20] In all complexes from previous work, it was performed single point calculation of the Merz-Singh-Kollman (MK) atomic charge[21] using M052X/6-311 + + G(2d,2p) level of theory [22] in Gaussian 09 software.

The molecular complexes to build the features and the label data for the dataset used in the ML modeling of our web application were collected from our previous works using LPED [6, 7, 23] and from this work. We built the dataset with one categorical feature which identifies the interacting atoms and their corresponding molecules involved in the complex. However, this categorical feature is dropped before performing the ML modeling due to its redundancy and to avoid potential overfitting. The dataset comprises three numerical features: interatomic distance (in Angstroms) and atomic charges (in MK or similar electrostatic potential schemes such as RESP and ChelpG) of the interacting atoms, with each atom belonging to a different molecule within the complex. These features were normalized to ensure consistent scales for ML algorithms. Furthermore, the target variable, the LPED, was derived from our QTAIM calculations conducted in previous studies for the previously examined complexes, as well as from the current study for the newly added complexes.

We performed an analysis of the optimal dataset analysis from 33, 43, 53, and 63 samples and found the best metrics for this simple dataset of only three numerical features with 53 samples. The optimized size of the dataset was sufficiently diverse to capture a wide range of intermolecular interactions and scenarios, enabling us to achieve a good accuracy in both primary and secondary testing for the best-tested model. The 53-sample dataset was used to train the ML models in primary testing in the first code and is used to train the best ML model in the second code, which is implemented in Flask framework.

Whenever possible, the atomic charge of the most electronegative interacting atom (the one with a negative sign) was always placed in the same numerical feature (Atomic charge A) in the dataset. In the web application, we recommend the user adopt the same procedure in order to avoid poorer predictive results, as confirmed in our preliminary testing.

From primary testing, we found that Linear Regression was the best model for the dataset since it had the best metrics. To further analyze the accuracy of the best-performing ML, we constructed a second dataset as input data. Geometry optimization, QTAIM, and MK analyses were performed on the complexes within this second dataset, resulting in a CSV file with the same features as the initial dataset which comprised the input data for the second testing. This file was then inputted into our web application, generating predicted LPED values for the best model. Finally, we used a spreadsheet to obtain the same metrics from primary testing to further validate the best model.

METHODOLOGY FOR THE CODE

The web application was structured using the Flask framework. The main libraries for ML modeling were Pandas, used for data manipulation and preprocessing, and Scikit-Learn, which provided the tools for machine learning algorithms. Visual Studio Code was utilized as the Integrated Development Environment (IDE) for coding and managing the project. The web application was deployed on PythonAnywhere's web hosting platform.

There are two separate Python codes for this web application. Indeed, only the second one was implemented in the Flask framework. The first code was not deployed and was used to test and obtain the metrics of the five ML models in order to choose the best one according to our dataset (so called primary testing) and to validate it using input data in secondary testing. The metrics used in the code were: coefficient of determination (R²), mean absolute error (MAE), mean squared error (MSE), and root mean squared error (RMSE).

To prevent data leakage, we normalized the training data (X_train and y_train) using the fit_transform() method. The same normalization was then applied to the test data (X_test) using the transform() method, which avoids introducing information from the test set into the training process. This procedure was realized in both codes.

In both codes, we applied data wrangling regarding the interatomic distance feature. It was transformed into Lennard-Jones potential according to the equation described in the code:

data['Lennard_Jones potential] = 4 * (

(data['Inverse interatomic distance']) ** 12 - (data['Inverse interatomic distance']) ** 6)

The Lennard-Jones potential provided better metrics for the ML models in comparison with interatomic distance.

We did not perform comprehensive hyperparameter tuning on the ML models because preliminary testing indicated that it did not significantly improve the models’ metrics. On the other hand, during primary testing, we explored the effect of varying the random seeds, setting it to values ranging from 0 to 9 for all models. This variation significantly impacted the performance metrics, highlighting the sensitivity of machine learning algorithms, even for models like linear regression, to random initialization. This observation underscores the importance of considering initialization strategies in model training.[24]

We also evaluated the influence of the test size in all ML models, ranging from 0.2 to 0.3, and we found that 0.2 provided the best metrics for all models.

Initial parameter optimization, including random seed and test size, was conducted using a train/test split analysis on the first dataset. This process identified the best ML model as being Linear Regression with optimal initial parameters that exhibited the best performance metrics. This model was then evaluated in secondary testing using the web application, generating predicted LPED values for the second dataset. The second testing gave close values of MAE and RSME compared with those from primary testing.

The second code, implemented within the Flask framework, utilizes the best-performing model identified in both primary and secondary testing. The resulting web application provides two distinct methods for data input:

Individual Value Input: Users can manually input individual values for interatomic distance and atomic charges of the interacting atoms.

CSV File Upload: Alternatively, users can upload a CSV file containing multiple rows, each representing an interaction. Each row must include the interatomic distance and atomic charges of the interacting atoms, along with a categorical feature identifying the specific interaction type."

Both alternatives also provide the corresponding local SME value according to the excellent correlation between SME and LPED[7]. Indeed, it is the local SME and global SME the most important property to be extracted from intra/intermolecular interactions.

Figure 1 displays the flow chart of the second code, implemented in Flask.

RATIONALE FOR THE USE OF LENNARD-JONES POTENTIAL IN DATA WRANGLING

The Lennard-Jones potential[25] is a simplified yet effective model that captures the essential features of interactions between simple atoms and molecules. Specifically, it describes how two interacting particles repel each other at very close distances, attract each other at moderate distances, and eventually cease to interact at infinite distances, as depicted in the accompanying figure. The Lennard-Jones potential is a pair potential, meaning it only considers interactions between two particles at a time and does not account for three-body or multi-body interactions. The functional form of Lennard-Jones potential typically includes a term proportional to $\:{r}^{-6}$, accounting for the attractive London dispersion forces, and a repulsive term, proportional to $\:{r}^{-12}$, that prevents atoms from collapsing into each other at short distances. This dual-component structure accurately reflects the physical reality of atomic interactions, balancing long-range attraction with short-range repulsion.

We have optimized the geometry of complexes (or individual molecule as in the case of t-DCTN) 1 to 15 whose data (interatomic distance and atomic charges) were used to complement the dataset for primary testing and for both codes, which also includes data from the complexes previously studied in our other works on LPED. [6, 7, 23]

From their optimized geometries, we have analyzed their QTAIM properties (used to obtain their corresponding local potential energy density) by means of their molecular graphs, which are displayed in Fig. 2. The element symbols indicated in this figure represent the interacting atoms of these complexes or individual molecule (8). The bond paths in the molecular graphs indicate chemical bonds and intra/intermolecular interactions. A pair of bond paths are united by a bond critical point (the smaller red point in the molecular graph).

The interactions in 8 (t-DCTN) are intramolecular, and they are responsible for its most stable conformation, which was already analyzed by experimental and theoretical techniques.[26] A couple of these intramolecular interactions (hydrogen-hydrogen bonding) are less known in chemical books, but they have been widely explored in the literature recently.[27, 28] Hydrogen-hydrogen bonds are interactions between hydrogen atoms bonded to carbon atoms (CH—HC).

Complex 11 was taken from the CYP17-Pregnenolone complex, which underwent classical molecular dynamics, docking, and ONIOM calculations. From this optimized complex, two selected CYP17 amino acids interacting with pregnenolone and pregnenolone itself were isolated from the rest of the enzyme, and their QTAIM data were collected. Then, complex 11 is made up of pregnenolone, arginine, and asparagine (from CYP17). In this complex, there are hydrogen bonds, hydrogen-hydrogen bonds, and a usual H-H bond. [29]

Table 1 shows all the parameters used in LPED calculation: the atomic charge of the pair of interacting atomic basins, q(A) and q(B), where A and B are atomic basins identified in each entry of Table 1. Important to note that atomic basin A always corresponds to the most electronegative interacting atomic basins containing the negative atomic charge. Table 1 also contains the localization index of the interacting atomic basins, LI(A) and LI(B); the effective atomic number of the interacting atoms, Z_eff(A) and Z_eff(B); the charge density of the bond critical point connected by the pair of bond paths to both atomic basins A and B, ρ_bcp; and the bond path length of both bond paths of the interaction or bond involving the atomic basins A and B, r_(1−bcp) and r_(2−bcp). The LPED of the complexes or dimers 1 to 21 is given in atomic units (au.) and in kcal mol^− 1 Bohr^− 3. All these values depicted in Table 1 were obtained from the topological analysis of the ωB97X-D[17]/6-311 + + G(2d,2p) wave function.

Table 1

QTAIM atomic charge, q, localization index, LI, effective atomic number, Z_eff, of the interacting atomics basins A and B, in au., charge density of the bond critical point, ρ_bcp, of inter/intramolecular interaction, in au.; bond path length, in Bohr unit, from atomic basin A to bcp, r (A-bcp), and from atomic basin B to bcp, r (B-bcp); local potential energy density, LPED, in au. and in kcal mol^− 1 Bohr^− 3, of all studied dimers and complexes (1 to 21) from ωB97X-D[17]/6-311 + + G(2d,2p) level of theory. The atomic basin A whenever possible corresponds to the most electronegative atom of the interacting pair.
Entry	q(A) / au.	LI(A) / au.	Z_eff(A) / au.	q(B) / au.	LI(B) / au.	Z_eff(B) / au.	ρ_bcp / au.	r_(A−bcp) /Bohr	r_(B−bcp) /Bohr	LPED / au.	LPED ^(a)
1^(b)	-1.280	8.420	0.860	0.915	9.960	0.125	-0.0290	2.320	1.8700	-0.0063	-3.98
2^(c)	-1.067	6.593	1.474	0.372	0.162	0.466	-0.0170	2.630	1.5810	-0.0073	-4.56
3^(d1)	-0.584	5.946	1.638	0.62	0.058	0.322	-0.0280	2.429	1.3600	-0.0128	-8.00
3^(d2)	-0.789	7.698	1.091	0.606	0.065	0.329	-0.0150	2.590	1.6560	-0.0046	-2.92
4^(e)	-0.961	6.464	1.497	0.025	16.161	0.814	-0.0296	2.450	2.4300	-0.0140	-8.79
5^(f)	-0.012	6.827	1.185	0.25	0.224	0.526	-0.0165	2.574	1.5260	-0.0066	-4.17
6^(g)	-1.133	8.175	0.958	0.613	0.059	0.328	-0.0238	2.316	1.4830	-0.0076	-4.74
7^(h1)	-1.098	8.151	0.947	1.050	3.271	1.679	-0.014	2.654	2.4690	-0.0073	-4.55
7^(h2)	-1.121	8.202	0.919	0.058	0.400	0.542	-0.0110	2.677	1.9740	-0.0034	-2.13
8⁽ⁱ¹⁾	-1.148	8.188	0.960	0.042	0.394	0.564	-0.0110	2.744	1.9900	-0.0035	-2.19
8⁽ⁱ²⁾	-0.019	0.449	0.570	0.022	0.421	0.557	-0.0079	2.283	2.3630	-0.0019	-1.20
9^(j)	0.018	16.238	0.744	0.001	0.437	0.563	-0.004	3.610	2.320	-0.0009	-0.56
10^(k1)	-0.323	16.643	0.680	0.893	9.967	0.140	-0.012	3.147	2.156	-0.0017	-1.06
10^(k2)	-0.877	17.681	0.196	0.133	0.318	0.549	-0.014	3.133	1.713	-0.0027	-1.68
11^(l1)	-1.074	6.453	1.621	0.032	0.400	0.568	-0.008	3.074	2.725	-0.0029	-1.80
11^(l2)	0.422	0.146	0.432	-0.026	0.458	0.568	-0.003	2.624	2.725	-0.0006	-0.39
11^(l3)	-1.134	8.132	1.002	0.537	0.085	0.378	-0.023	2.357	1.390	-0.0080	-5.03
11^(l4)	-1.134	8.132	1.002	0.032	0.400	0.568	-0.008	3.192	2.127	-0.0022	-1.37
11^(l5)	0.414	0.143	0.443	0.036	0.398	0.566	-0.008	2.417	2.620	-0.0015	-0.96
11^(l6)	-1.225	6.635	1.590	0.049	0.392	0.559	-0.008	2.919	2.553	-0.0030	-1.87
11^(l7)	-0.895	7.785	1.110	0.477	0.110	0.413	-0.018	2.511	1.482	-0.0065	-4.07
11^(l8)	-0.895	7.785	1.110	0.056	0.383	0.561	-0.008	2.876	1.998	-0.0025	-1.59
12^(m)	-1.126	8.177	0.949	0.598	0.064	0.338	-0.025	2.381	1.323	-0.0082	-5.15
12^(m)	-1.158	8.432	0.726	0.069	0.379	0.552	-0.009	2.722	1.968	-0.0024	-1.51
13⁽ⁿ⁾	-0.067	6.646	1.421	0.010	3.932	2.058	-0.013	2.446	2.871	-0.0084	-5.25
14^(o)	-1.131	8.162	0.969	0.641	0.051	0.308	-0.047	2.155	1.130	-0.0169	-10.58
15^(p1)	-0.033	3.966	2.067	0.020	0.411	0.569	-0.006	3.191	2.287	-0.0026	-1.66
15^(p2)	0.020	0.411	0.569	0.020	0.408	0.572	-0.011	2.127	2.126	-0.0029	-1.80
15^(p3)	-0.005	3.975	2.030	0.0104	3.963	2.027	-0.004	3.449	3.706	-0.0024	-1.53
16^(q)	-1.208	6.875	1.333	0.597	0.065	0.338	-0.0195	2.532	1.415	-0.0075	-4.68
17^(r)	-1.167	8.422	0.745	0.535	0.089	0.376	-0.0450	2.136	1.095	-0.0156	-9.77
18^(s1)	-1.096	7.899	1.197	0.560	0.077	0.363	-0.0180	2.493	1.470	-0.0065	-4.11
18^(s2)	-1.167	8.217	0.950	0.607	0.061	0.332	-0.0250	2.365	1.314	-0.0082	-5.13
19^(t1)	-1.177	8.238	0.939	0.078	0.369	0.553	-0.0082	2.818	2.062	-0.0025	-1.55
19^(t2)	1.557	2.822	1.621	1.559	2.820	1.621	-0.0054	3.027	3.027	-0.0029	-1.81
19^(t3)	-1.087	7.905	1.182	-1.083	7.899	1.184	-0.0076	2.877	2.877	-0.0031	-1.96
20^(u)	-1.092	7.898	1.194	0.601	0.062	0.337	-0.0293	2.320	1.254	-0.0115	-7.20
21^(v1)	-1.154	8.185	0.969	0.598	0.064	0.338	-0.0243	2.363	1.322	-0.0081	-5.08
21^(v2)	-1.152	8.429	0.723	0.032	0.411	0.557	-0.0068	2.929	2.254	-0.0017	-1.05
(a) Unit in kcal mol^-1 Bohr^-3;
(b) O-Na interaction;
(c) N-H interaction;
(d1) N-H interaction; (d2) O-H interaction;
(e) N-Cl interaction;
(f) O-H interaction;
(g) O-H interaction;
(h1) O-C interaction; (h2) O-HC interaction;
(i1) O-HC interaction; (i2) H-H interaction;
(j) Cl-H interaction;
(k1) Cl-Na interaction; (k2) Cl-HC interaction;

(l1) N-H (ASN-PREG) interaction; (l2) H-H (ASN-PREG) interaction; (l3) O-H (ASN-PREG) interaction; (l4) O-H (ASN-PREG) interaction; (l5) NH-H (ARG-PREG) interaction; (l6) N-H (ARG-PREG) interaction; (l7) O-HN (PREG-ARG) interaction; (l8) O-H (PREG-ARG) interaction;

(m) O-H interaction;
(n) O-C interaction;
(o) O-H interaction;

(p1) C-H interaction; (p2) H-H interaction; (p3) C-C interaction;

(q) N-H interaction;
(r) O-H interaction;

(s1) O(sp³)-H interaction ; (s2) O(sp²)-H interaction;

(t1) O(sp²)-H interaction; (t2) O(sp²)-O(sp²) interaction; (t3) O(sp³)-O(sp³) interaction;

(u) O-H interaction;

(v1) O-H interaction; (v2) O-H(C) interaction.

The dataset for training the ML models in the primary testing from the first code and for training the best ML model in the code implemented in Flask framework is depicted in Table 2. Each row is identified by index. The intra/intermolecular interactions in each row are represented in the Interaction column (or variable). The interatomic distance is in Angstrom units. The atomic charges in the interacting atoms A and B were obtained from the MK scheme. The LPED column (or variable) has the actual values of each interaction to be used in the train/test split and predictive algorithm.

The complexes from which we had previously their optimized geometries and LPED values [5–7] are indicated in the “Interaction” variable from index 1 to 24. The remaining index values (from 25 to 53) correspond to the complexes included in this work. In the “Interaction” categorical feature used before data wrangling, in some rows, ‘avg’ appears representing the averaged values (regarding the numerical features) from similar interactions in the corresponding complex in these rows. It is worth noting that interatomic distance and atomic charges from MK scheme are geometrical and electrostatic potential properties, respectively, and they do not have any physical relation with the topological data to obtain LPED (depicted in Table 1).

Table 2

Dataset for training the ML models in both web application codes. Interatomic distances are in the Angstrom unit. Atomic charges were obtained from the MK scheme. LPED unit is kcal mol^− 1 Bohr^− 3.
Index	Interaction	Interatomic distance	Atomic charge A (MK)	Atomic charge B (MK)	LPED
1	Cl-Ar (HCl-Ar)	3.976	-0.212	0.001	-0.18
2	Cl-H (chloromethane dimer) (avg)^(a)	3.121	-0.165	0.154	-0.88
3	F-F (tetrafluoromethane dimer) (avg)	3.063	-0.210	-0.184	-0.53
4	O-H (methyl ether dimer) (avg)	2.771	-0.295	0.087	-1.17
5	O-O (methyl ether dimer)	3.220	-0.295	-0.271	-1.42
6	O-Cl (formaldehyde-Cl₂)	3.304	-0.439	0.004	-0.80
7	O-H (water-ethyne)	2.179	-0.824	0.410	-2.48
8	H-H (propane dimer) (avg)	2.636	0.083	0.082	-0.46
9	C-C (benzene dimer) (avg)	3.529	-0.166	-0.083	-2.07
10	O-H (nitrophenol)	1.723	-0.471	0.461	-10.93
11	O-H (formic acid dimer)	1.634	-0.653	0.571	-11.11
12	O-H (formamide dimer)	1.987	-0.539	0.350	-4.56
13	O-H (methanol dimer)	1.921	-0.582	0.391	-5.24
14	N-H (methylamine dimer)	2.247	-0.818	0.301	-4.21
15	S-H (methanethiol dimer)	2.852	-0.301	0.165	-2.07
16	O-H (water dimer)	1.916	-0.771	0.396	-3.72
17	N-H (ammonia dimer)	2.317	-0.777	0.264	-3.16
18	F-H (HF dimer)	1.811	-0.402	0.392	-2.43
19	S-H (H₂S dimer)	2.845	-0.233	0.118	-1.82
20	O-Ca (Fosfomycin-Ca^+ 2) (avg)	2.209	-0.780	1.475	-13.18
21	O-H (guanine-cytosine) (avg)	1.820	-0.715	0.629	-7.83
22	N-H (guanine-cytosine)	1.891	-0.931	0.484	-10.95
23	O-Ca (EDTA-Ca^+ 2) (avg)	2.325	-0.957	1.497	-8.86
24	N-Ca (EDTA-Ca^+ 2) (avg)	2.511	0.531	1.497	-10.38
25	O-Na (sodium-formate) (avg)	2.223	-0.776	0.892	-6.60
26	N-H (methanimine dimer)	2.202	-0.652	0.343	-4.56
27	N-H (methanoxime-dimer)	1.972	-0.101	0.277	-8.00
28	O-H (methanoxime-dimer)	2.196	-0.326	0.393	-2.92
29	O-HC (DCTN)	2.456	-0.561	0.076	-2.19
30	H-H (DCTN)	2.285	0.119	0.086	-1.20
31	N-Cl (ammonia-chlorine)	2.582	-0.344	-0.155	-8.79
32	O-H (hydroxy-acetaldehyde - dimer)	1.898	-0.416	0.358	-4.74
33	O-H (oxygen-hydrogen chloride)	2.149	0.055	0.147	-4.17
34	O-C (formaldehyde-dimer)	2.708	-0.431	0.315	-4.55
35	O-HC (formaldehyde-dimer)	2.417	-0.427	0.016	-2.13
36	N-H (ASN-Preg)^(b)	2.707	-1.365	-0.072	-1.80
37	H-H (ASN-Preg)	2.599	-0.065	0.494	-0.39
38	O-HO (ASN-Preg)	1.954	-0.668	0.511	-5.03
39	O-HC (ASN-Preg)	2.704	-0.668	-0.072	-1.37
40	NH-H (ARG-Preg)	2.222	0.004	0.518	-0.96
41	N-H (ARG-Preg)^(c)	2.726	-1.626	0.067	-1.87
42	O-HN (Preg-ARG)	2.089	-0.947	0.695	-4.07
43	O-H (Preg-ARG)	2.549	-0.947	0.103	-1.59
44	Cl-H (trimethylamine-chlorine)	3.120	-0.002	0.154	-1.87
45	Cl-Na (methyl chloride-NaCl)	2.810	-0.120	0.765	-4.07
46	Cl-H (NaCl-methyl chloride)	2.540	-0.785	0.000	-1.59
47	O-H (butanal-water)	1.935	-0.465	0.359	-5.15
48	O-H (water-butanal)	2.455	-0.828	0.060	-1.51
49	O-C (⁺NO₂-benzene) (avg)	2.801	0.367	-0.224	-5.25
50	O-H (β-hydroxy acrolein)	1.713	-0.631	0.546	-10.58
51	C-H (cis-butene-dimer)	2.862	-0.219	0.001	-1.66
52	H-H (cis-butene-dimer)	2.090	0.094	0.001	-1.80
53	C-C (cis-butene-dimer)	3.510	-0.013	0.010	-1.53
(a) (avg) means averaged values
(b) ASN = asparagine from CYP17, Preg = pregnenolone
(c) ARG = arginine from CYP17

We validated the optimal size of our dataset using three metrics: coefficient of determination, R², mean absolute error, MAE, and root mean squared error, RMSE. We tested the dataset size with 33, 43, 53, and 63 samples for the best ML model, Linear Regression (see ahead), in their corresponding optimal random_state values (see Table 3). Bouasria et al. have done a similar analysis in a much more complex dataset with 29 numerical features for 25, 50, 100, 200, 300, and 500 samples and found the best dataset size with 300 samples.[30] Our dataset is much simpler, having only 3 numerical features and its optimal size has 53 samples. The rule of thumb in linear regression and logistic regression models is a minimum of 10–20 samples for each feature[31, 32] and our optimal dataset is within the minimum statistically determined range.

Table 3

Metrics of the Linear Regression model (coefficient of determination, R², mean absolute error, MAE, in kcal mol^− 1 Bohr³, and root mean squared error, RMSE, in kcal mol^− 1 Bohr³) for different dataset sizes and their corresponding optimal random_state values.
Dataset size	Random_state	R²	MAE^(a)	RMSE^(a)
33	5	0.75	1.42	2.04
43	5	0.75	1.12	1.46
53	8	0.88	0.72	0.91
63	0	0.83	0.94	1.26
(a) In kcal mol^-1 Bohr³

The statistical analysis of this dataset after data cleaning and data wrangling is depicted in Table 4 where ‘count’ represents the total number of rows; ‘std’ means standard deviation; ‘min’ is the smallest value of each variable ‘25%’ is the first quartile that gives the value where the first quarter of the observations lie below this one, after ordering of the values from the smallest to the highest value; ‘50%’ (median) and ‘75%’ with similar concepts in comparison with ‘25%’ but applied to 50% and 75% of the observations below the values indicated in ‘median’ and ‘75%’, respectively; and ‘max’ is the largest value observed in each variable.

For the ‘Atomic charge A’ and ‘Atomic charge B’ columns, the mean and median (50%) are close, indicating a normal distribution and low dispersion of values. However, the mean and median in ‘Lennard_Jones potential’ and LPED columns are not close, indicating a non-normal distribution and more dispersed values. In Fig. 4, the histogram plots confirm the normal distribution of ‘Atomic charge A’ and ‘Atomic charge B’ features and the non-normal distribution of the Lennard_Jones potential and LPED variables.

Table 4

Statistical analysis of the dataset used in the web application.
	Atomic charge A (MK)	Atomic charge B (MK)	Lennard_Jones potential	LPED
count	53	53	53	53
mean	-0.447	0.299	-0.0382	-4.01
std	0.409	0.392	0.0430	3.37
min	-1.626	-0.271	-0.1991	-13.18
25%	-0.771	0.016	-0.0476	-5.15
50%	-0.431	0.264	-0.0200	-2.48
75%	-0.166	0.461	-0.0088	-1.59
max	0.531	1.497	-0.0010	-0.18

In Fig. 5, the edges of the box represent the first quartile (Q1) and third quartile (Q3) of the interquartile range (IQR). The extremes delimited are the upper and lower whiskers, indicating variability outside the upper and lower quartiles. The variables ‘Atomic charge A’, ‘Atomic charge B’, and ‘Lennard_Jones potential’ have zero, two, and five outlier(s), respectively. However, they are mild outliers because they are close to the upper or lower whiskers. Since they are mild outliers, they were not removed.

The correlation measures the strength and direction of a linear relationship between two variables. In Table 5, the correlation values between variables are shown. The feature ‘Atomic charge B’ (where the great majority of positive atomic charges are placed) has a negative and highest (in magnitude) correlation with LPED, followed by Lennard_Jones potential and ‘Atomic charge A’ variables with positive correlations.

Table 5

Correlation values between the variables ‘Atomic charge A’, ‘Atomic charge B’, and ‘Lennard_Jones potential’
	Atomic charge A (MK)	Atomic charge B (MK)	Lennard_Jones potential	LPED
Atomic charge A (MK)	1.00	-0.12	0.20	0.15
Atomic charge B (MK)	-0.12	1.00	-0.32	-0.66
Lennard_Jones potential	0.20	-0.32	1.00	0.62
LPED	0.15	-0.66	0.62	1.00

In the primary testing in our first code, we searched for the best ML model and the best random seed varying random_state value from 0 to 9 (called iteration). The metrics used to evaluate the ML model are R², MAE, MSE, and RMSE. According to the metrics depicted in Table 6, the best model in the best random seed value is Linear Regression with random_state = 8 having R² = 0.88, MAE = 0.72 kcal mol^− 1 Bohr^− 3, MSE = 0.82 kcal² mol^− 2 Bohr^− 6, and RMSE = 0.91 kcal mol^− 1 Bohr^− 3.

Table 6

R², MAE, MSE, and RMSE metrics for the tested ML models according to the range 0–9 of random_state (iteration).
Model	Iteration	Split	R²	MAE^(a)	MSE^(b)	RMSE^(a)
Linear Regression	0	1	0.54	1.12	2.07	1.44
	1	2	0.45	1.92	9.31	3.05
	2	3	0.58	1.58	4.63	2.15
	3	4	-0.30	2.12	10.65	3.26
	4	5	0.33	1.62	4.46	2.11
	5	6	0.54	1.13	3.15	1.77
	6	7	0.56	1.77	5.32	2.31
	7	8	0.42	1.46	6.78	2.60
	8	9	0.88	0.72	0.82	0.91
	9	10	0.67	1.46	3.73	1.93
Decision Tree	0	1	-1.91	2.66	13.26	3.64
	1	2	-0.18	3.14	19.93	4.46
	2	3	0.07	2.60	10.20	3.19
	3	4	-1.14	3.03	17.48	4.18
	4	5	-1.13	2.89	14.18	3.77
	5	6	-1.11	2.76	14.47	3.80
	6	7	0.69	1.37	3.77	1.94
	7	8	0.35	1.62	7.53	2.74
	8	9	-0.73	2.50	11.47	3.39
	9	10	0.11	2.38	9.98	3.16
Random Forest	0	1	-2.07	2.85	13.98	3.74
	1	2	0.43	2.20	9.68	3.11
	2	3	0.55	1.88	5.01	2.24
	3	4	-0.23	2.36	10.10	3.18
	4	5	0.31	1.87	4.60	2.15
	5	6	0.32	1.68	4.65	2.16
	6	7	0.50	1.58	5.98	2.44
	7	8	0.56	1.26	5.10	2.26
	8	9	0.83	0.87	1.15	1.07
	9	10	0.53	1.75	5.27	2.30
XGBoost	0	1	-0.99	2.24	9.04	3.01
	1	2	0.20	2.64	13.45	3.67
	2	3	0.18	2.34	9.00	3.00
	3	4	-0.63	2.88	13.37	3.66
	4	5	-0.21	2.15	8.08	2.84
	5	6	-0.71	2.47	11.73	3.42
	6	7	0.58	1.34	4.99	2.23
	7	8	0.44	1.55	6.52	2.55
	8	9	0.64	1.27	2.39	1.54
	9	10	0.31	1.99	7.80	2.79
Gradient Boosting	0	1	-1.65	2.56	12.07	3.47
	1	2	0.46	2.07	9.18	3.03
	2	3	0.55	1.85	4.99	2.23
	3	4	-0.35	2.48	11.06	3.33
	4	5	-0.25	2.16	8.35	2.89
	5	6	-0.92	2.29	13.15	3.63
	6	7	0.74	1.23	3.15	1.78
	7	8	0.46	1.35	6.22	2.49
	8	9	0.80	0.96	1.31	1.15
	9	10	0.32	1.99	7.68	2.77
(a) kcal mol^-1 Bohr^-3
(b) kcal² mol^-2 Bohr^-6

The regression equation of LPED from the interatomic distance and electrostatic potential atomic charges is:

y = -1.5993 + (-0.2092 *x1) + (-4.3116 *x2) + (35.4112 *x3)

Where y = LPED, x1 = Atomic charge A (MK), x2 = Atomic charge B (MK), and x3 = Lennard_Jones potential.

After finding the best model in the best random seed, we applied secondary testing to validate the model further. We used complexes 16 to 21 (Fig. 3) to obtain the MK atomic charges of their interacting atoms, their corresponding interatomic distance in Angstroms, and their corresponding LPED values from their topological data (see Table 1), which we compared with the predicted LPED. The input data for the secondary testing has the “complex” categorical variable and the numerical variables ‘Interatomic distance’, ‘Atomic charge A’, and ‘Atomic charge B’, where most of the values in ‘Atomic charge A’ are negative and most values in ‘Atomic charge B’ (see Table 7). This table, in xlsx extension, was changed into csv file to be uploaded in our web application [10].

Table 7

Input data for the secondary testing.
Complex	Interatomic distance	Atomic charge A	Atomic charge B
N-H (propanenitrile-water)	2.063	-0.440	0.382
O-H (methylammonium-water)	1.688	-1.039	0.412
O^(a)-H (methyl acetate-water)	2.070	-0.136	0.262
O^(b)-H (methyl acetate-water)	1.920	-0.550	0.331
O^(b)-H (methyl acetate-dimer) (avg)	2.543	-0.584	0.132
C^(b)-C^(b) (methyl acetate-dimer)	3.172	0.820	0.830
O^(a)-O^(a) (methyl acetate-dimer)	3.043	-0.377	-0.382
O-H (diethyl ether - water)	1.866	-0.422	0.404
O-H (2-pentanone-water)	1.922	-0.564	0.367
O-HC (2-pentanone-water)	2.642	-0.870	0.013
(a) Tetrahedral oxygen
(b) Trigonal oxygen

After uploading and obtaining the prediction values, we downloaded a CSV file called ‘predictions.’ We changed the extension of this file back to xlsx. The ‘predictions’ file has two additional columns: Predicted_LPED and Predicted_SME. We compared the values of the Predicted_LPED with their corresponding actual LPED values and applied the same metrics of the primary testing. The results are displayed in Table 8. For the secondary testing, we computed the metrics using a spreadsheet. Firstly, we obtained the absolute and squared differences between actual and predicted LPED values, represented by Abs_diff and Sqr_diff columns, respectively, in Table 7. After that, we obtained MAE (1.19 kcal mol^− 1 Bohr^− 3), MSE (1.59 kcal² mol^− 2 Bohr^− 6), and RMSE (1.26 kcal mol^− 1 Bohr^− 3). These values are moderately close to those from the primary testing (MAE = 0.72 kcal mol^− 1 Bohr^− 3, MSE = 0.82 kcal² mol^− 2 Bohr^− 6, and RMSE = 0.91 kcal mol^− 1 Bohr^− 3.

Then, we could validate the Linear Regression model with random_state = 8 as the best ML model for predicting the LPED values of intermolecular interactions in complexes or intramolecular interactions in individual molecules and, consequently, obtain their corresponding local supramolecular energy (SME) from the linear relation between LPED and SME obtained in previous work[7].

If there is more than one local SME, the total SME of the complex is the sum of all local SMEs. This web application represents a simpler and easier way to obtain local SME of complex with multiple interactions without resorting to QTAIM calculations, and it shows the utility of machine learning modeling in obtaining a topologically derived information, such as LPED, using non-topological data.

Table 8

Predicted and actual values of LPED along with their corresponding absolute and squared differences for the input data of the secondary testing.
Predicted_LPED	Actual_LPED	Abs_diff	Sqr_diff
-4.97	-4.68	0.29	0.08
-9.02	-9.77	0.75	0.57
-4.48	-4.11	0.37	0.14
-5.68	-5.13	0.55	0.30
-2.57	-1.55	1.02	1.05
-5.49	-1.81	3.68	13.51
-0.05	-1.96	1.91	3.65
-6.53	-7.20	0.67	0.45
-5.82	-5.08	0.74	0.55
-1.89	-1.05	0.84	0.70

We developed a Flask web application, hosted on PythonAnywhere, to predict local potential energy density (LPED) and corresponding local supramolecular energy (SME) for both intermolecular and intramolecular interactions using machine learning from an optimized size of the dataset. This application simplifies the calculation by using atomic charges (determined by MK, ChelpG, or RESP schemes) and interatomic distances in Angstroms, bypassing the complexity of QTAIM analysis.

Machine learning modeling plays an important role in building a bridge between simple inputs (geometric and electrostatic potential data) and topological data (charge density of bond critical points, bond paths, and effective atomic numbers from localization index and Bader’s atomic charge) to obtain LPED.

We evaluated five different machine learning models under various random seed scenarios and identified linear regression as the model with the best performance. The metrics for this model, confirmed by a secondary validation against known LPED data, showed close values for MAE and RMSE, validating the effectiveness of linear regression in predicting LPED and SME, consequently.

CONFLICTS OF INTEREST

The authors declare that they have no conflicts of interest.

DATA AVAILABILITY

The Python code for the analysis of the best machine learning model is available at: https://doi.org/10.5281/zenodo.12752491

The Flask web application is available at:

https://clfirme.pythonanywhere.com/

FUNDING DECLARATION

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Local potential energy density-supramolecular energy (LPED-SME) machine learning prediction – a web application to obtain the local SME from simple inputs

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Abstract

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INTRODUCTION

RATIONALE ON THE TESTED MACHINE LEARNING MODELS

COMPUTATIONAL DETAILS FOR BUILDING THE DATASET

METHODOLOGY FOR THE BUILDING THE DATASET AND THE SECONDARY TESTING

METHODOLOGY FOR THE CODE

RATIONALE FOR THE USE OF LENNARD-JONES POTENTIAL IN DATA WRANGLING

RESULTS AND DISCUSSION

CONCLUSIONS

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References

Additional Declarations

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