Machine learning analysis of molecular dynamics properties influencing drug solubility

Abstract

Solubility is critical in drug discovery and development, as it significantly influences a medication’s bioavailability and therapeutic efficacy. Understanding solubility at the early stages of drug discovery is essential for minimizing resource consumption and enhancing the likelihood of clinical success via prioritizing compounds with optimal solubility. Molecular dynamics (MD) simulation is a powerful computational tool for modeling various physicochemical properties, particularly solubility. MD simulations offer a detailed perspective on molecular interactions and dynamics, providing insights into the factors influencing solubility. This study aims to statistically examine the impact of ten MD-derived properties, along with octanol-water partition coefficient (logP), one of the most influential experimental properties, on the aqueous solubility of drugs using Machine Learning (ML) techniques. To achieve this, a dataset comprising 211 drugs from diverse classes was compiled from the literature. These drugs were subjected to MD simulation, from which relevant properties were extracted and selected as features. Additionally, the corresponding logP from previous studies was incorporated into the analysis. Through rigorous analysis, the properties with the most significant influence on solubility were identified and subsequently used as input features for four ensemble machine learning algorithms: Random Forest, Extra Trees, XGBoost, and Gradient Boosting. The results indicate that seven properties, logP, Solvent Accessible Surface Area (SASA), Coulombic_t, LJ, Estimated Solvation Free energies (DGSolv), Root Mean Square Deviation (RMSD), and Average number of solvents in Solvation Shell (AvgShell) are highly effective in predicting solubility, exhibiting performance comparable to predictive models based on structural features. The Gradient Boosting algorithm achieved the best performance with a predictive R² of 0.87 and an RMSE of 0.537 in test set. This research underscores the potential of integrating MD simulations with ML methodologies to improve the accuracy and efficiency of aqueous solubility predictions in drug development.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-11392-1.

Introduction

The aqueous solubility of a chemical compound is a pivotal physicochemical property in drug discovery and development¹. Recent evaluations have highlighted that many newly nominated chemical entities fail to reach the market due to their lipophilic characteristics and inadequate solubility². During the preliminary stages of drug development, assessing the aqueous solubility of potential pharmaceutical agents is imperative. This assessment is a practical strategy to minimize resource and time expenditures while mitigating the risk of failure by excluding compounds with suboptimal solubility profile.

Solubility exerts a profound influence on the pharmacokinetics and ADME-T (absorption, distribution, metabolism, excretion, and toxicity) characteristics of drug candidates, thereby determining their bioavailability through modulation of dissolution rates and the maximum concentration achievable within gastrointestinal fluids³. Insufficient solubility can pose substantial risks to patient safety, potentially leading to physical precipitation or crystallization of the compound^2,3. The application of Galenic formulation⁴, which involves preparing and combining medications to enhance their bodily absorption, can significantly improve a compound’s therapeutic efficacy. Consequently, selecting a drug candidate with favorable solubility considered prudent for advancing to clinical trials.

Diverse techniques are employed to enhance solubility, categorized into three main groups: Physical Modifications, Chemical Alterations, and Miscellaneous methods. Examples of these techniques include particle size reduction, nanosuspension, drug dispersion using surfactant, pH adjustment, co-crystallization, co-solvency, salt formation, supercritical fluid technology, micellar solubilization, direct capsule filling, and electrospinning, among others. Each approach has distinct advantages and limitations, necessitating careful consideration in their application^5,6.

In pharmaceutical research, solubility is typically elucidated through two distinct concepts: thermodynamic solubility and kinetic solubility. Thermodynamic solubility denotes the equilibrium concentration of a compound within a fully saturated solution, where an excess of undissolved solid persists. At this equilibrium, a dynamic balance is established between the solute dissolution rate and the recrystallization rate, resulting in a stable, maximum concentration. Conversely, kinetic solubility refers to the concentration at which a compound begins to precipitate from a supersaturated solution⁷.

Techniques such as the shake-flask and column elution methods are commonly employed to measure thermodynamic solubility^8,9. The shake-flask method involves agitating a hydrophilic solute in water until uniform distribution is achieved, followed by phase separation through centrifugation or filtration. In contrast, the column elution method entails passing water through a column coated with a solid chemical compound, maintaining a continuous flow at a constant rate until equilibrium is attained.

For the assessment of kinetic measurement, various analytical methods are utilized including: High-Performance Liquid Chromatography Ultraviolet (HPLC–UV)¹⁰, nephelometry¹⁰, turbidimetry¹¹, Quantitative Nuclear Magnetic Resonance (qNMR)¹², and Static Light Scattering (SLS)¹³, among others. These experimental approaches generally yield reliable and accurate results. However, their labor-intensive and resource-demanding procedures often lead to inefficiencies, material wastage, and environmental concerns¹⁴. Consequently, solubility prediction continues to be a dynamic and actively pursued area of research within the scientific community.

Over the past few decades, significant progress has been made in developing computational methods for predicting aqueous solubility, encompassing semi-empirical models, in silico Quantitative Structure–Property Relationship (QSPR) approaches, and advanced machine learning algorithms. Pioneering studies have explored the relationship between chemical structure and biological activity to identify suitable solvents that effectively approximate the aqueous and lipid phases of biological tissues¹⁵. Notably, Hansch et al. employed 1-octanol and water to determine the partition coefficient, which indicates the hydrophobic bonding constant. Their findings suggest that, except for hydrocarbons, comparable partitioning behavior can be anticipated from nearly any monofunctional liquid representing the lipid phase. These investigations established a linear correlation between the logarithm of aqueous solubility of organic liquids and their octanol–water partition coefficients¹⁵. Building on these insights, Valvani and Yalkowsky formulated the General Solubility Equation (GSE) through semi-empirical analysis. The GSE demonstrates that the logarithm of solubility (logS) is strongly correlated with the logarithm of the partition coefficient (logP), incorporating an additional term for the melting point (MP) to account for crystalline solutes¹⁶.

Solubility correlation with a range of structural and physical descriptors, including the Coulombic and Lennard–Jones interaction energies between solutes and water, the solvent-accessible surface area (SASA) along with its hydrophobic, hydrophilic, and aromatic subdivisions, the count of hydrogen bonds and LogP carried out by Monte Carlo (MC) simulations leads to the development of a regression model for logS that requires only five descriptors¹⁷.

Through diverse statistical modeling techniques, QSPR models have been practical computational tools that have exploited molecular structures to establish mathematical relationships among physicochemical properties, biological activities, and molecular descriptors^18,19. New ML techniques have transcended limitations, with improved handling of complex, high-dimensional data and complex descriptor interactions^20–24. ML-based QSPR uses descriptors like topological, electronic, and graph-based features. ML methods such as Multivariable linear regression (MLR), Artificial neural networks (ANN), Gaussian process (GP), and Support vector machine (SVM)^25–28 outperform linear models due to non-linear solubility-descriptor relationships. The study by Ye et al.²⁹ collected a dataset of 5081 experimental solubilities for small-molecule compounds in organic solvents. To encode structural features, molecular structures were represented using Extended-Connectivity Fingerprints (ECFPs). The authors compared nine ML algorithms for solubility prediction. Nonlinear ML models, including lightGBM, deep neural networks (DNN), support vector machines (SVM), random forest (RF), and extra trees (ET), outperformed linear models such as partial least squares (PLS) and Ridge regression. This enhanced performance is attributed to the intricate, nonlinear relationships between solubility and molecular structural attributes. The examination by Boobier et al.³⁰ stated that, unlike conventional QSAR/QSPR models that rely on descriptor sets (> 100), the authors rationally selected a limited set of 14 descriptors based on physicochemical aspects of the dissolution process, including lattice energy, solvation energy, and solute–solvent interactions. A recent comprehensive review by Llompart et al.³¹ systematically evaluated methods, datasets, descriptors, and predictive accuracies in solubility prediction studies published between 1997 and 2023. It addressed the challenge of aqueous solubility prediction by proposing standard data curation procedures and data-curated datasets to enhance machine learning models reliability and applicability in drug discovery.

This study aims to evaluate the efficacy of MD-derived properties in predicting aqueous solubility and to elucidate their comparative impact relative to other descriptors through the application of machine learning algorithms. The primary objective of the present research is to explore the influence of properties derived from molecular dynamics (MD) on the prediction of aqueous solubility, focusing on an improved understanding of their contribution to solubility modeling instead of the optimization of predictive performance alone.

In this context, an extensive collection of aqueous solubility data across a set of drug classes was assembled, and MD simulations were carried out to calculate associated physicochemical properties. logP was included as an additional feature due to its well-established significance in solubility studies. Statistical analysis through feature selection methods determined the importance of various solubility properties. The selected properties were subsequently used to construct predictive models using four ensemble machine learning algorithms: Random Forest (RF), Extra Trees (EXT), eXtreme Gradient Boosting (XGB), and Gradient Boosting Regression (GBR). These models were developed to predict solubility and compare the performance of MD-derived properties with previous results from predictive models based on structural descriptors. The findings show that MD-derived properties possess comparable predictive power, bringing to the fore their substantial impact and worth in encouraging better aqueous solubility predictions.

Methodology

Data collection

The target variable in this study is the logarithmic measure of aqueous solubility (logS) of compounds present in moles per liter (mol/L), ranging from − 5.82 (thioridazine) to 0.54 (ethambutol). This dataset, derived from the comprehensive study by Huuskonen et al³², encompasses experimental solubility values for 211 drugs and related compounds. Its selection was based on its extensive coverage and consistent experimental data, which is a prime requirement for robust training and validation of ML models. The dataset includes diverse drug classes, with their respective contributions illustrated in Fig. 1.

Fig. 1.

This pie chart illustrates the percentage distribution of drugs and derivatives across various classes: barbiturates (25%), steroids (16%), pteridines (14%), NSAIDs (8%), pyrimidines (8%), xanthine (7%), RTIs (6%), phenothiazine (4%), and “others” (12%).

In addition to MD-derived properties, the compounds’ octanol/water partition coefficient (logP) was incorporated as a critical feature extracted from scholarly literature, which is detailed in Table S4 of the supplementary materials. The inclusion of logP facilitates a comparative analysis between MD-derived properties and a well-established experimental descriptor, widely recognized for its significant influence on aqueous solubility, thereby enabling an evaluation of the efficacy of MD-derived properties. From 211 drugs in the Huuskonen et al.³² dataset, 12 Reverse-Transcriptase Inhibitors (RTIs) were excluded since the literature did not contain trustworthy logP values available. Including compounds with missing or untrustworthy logP values would be prejudicial to data integrity and adversely affect model performance. ML methods’ performance is highly dependent on having completeand accurate features, indicating the imperative of this exclusion.

MD simulations setup

Molecular dynamics simulations were conducted in the isothermal-isobaric (NPT) ensemble using the GROMACS 5.1.1 software package ³³. The GROMOS 54a7 force field was employed to model the molecules’ neutral conformation, generating topology and initial coordinate files^34,35. Simulations were performed within a cubic simulation box of dimensions (4 × 4 × 4) nm³, with periodic boundary conditions applied in all three spatial dimensions to ensure continuity of system properties. A variable number of water molecules modeled using the Extended Simple Point Charge (SPC/E) model were incorporated to hydrate the system.

The steepest descent energy minimization technique was applied to eliminate unfavorable atomic interactions and stabilize water molecules within the systems³⁶. Drug molecule positions were constrained during the preparatory phase, and simulations were conducted for 10 ns to achieve equilibrium. Subsequently, simulations proceeded for an additional 20 ns in the NPT ensemble without positional constraints. Thermal conditions were maintained at 300 K using the V-rescale thermostat with a time constant of 0.1 ps³⁷, while pressure was regulated at 1 bar, using the Parrinello-Rahman barostat with a compressibility factor of (4.5 × 10^–5) bar ^-1 and a time constant of 2 ps³⁸. The leap-frog algorithm was employed to integrate Newton’s equations of motion³⁹. Long-range electrostatic interactions were computed using the Particle Mesh Ewald (PME) method⁴⁰, with a cutoff radius of 1 nm set for both Coulombic and van der Waals interactions. Given the large number of compounds analyzed, results of the MD simulations for a representative compound are presented in the supplementary materials (Figs. 4S–8S).

Feature extraction

The analysis of molecular interactions during the dissolution of a compound in a solvent is essential for elucidating solubility characteristics. Based on this principle, molecular descriptors contain three energetic terms: lattice energy associated with crystal packing, the energy required to form a cavity in the solvent, and solvation energy, which collectively facilitate the modeling of aqueous solubility⁴¹. In this study, properties derived from molecular dynamics simulation were extracted to assess their impact on examining the aqueous solubility of drugs. To this end, key features were categorized into four aspects.

1. Structural properties Solvent Accessible Surface Area (SASA), Solvent Orientation around solute (Sorient), and Average number of solvents in Solvation Shell (AvgShell)

2. Intermolecular properties Average Number of Hydrogen Bonds (Hbond), Coulombic and Lennard–Jones (LJ) Energies

3. Dynamic property Root Mean Square Deviation (RMSD)

4. Thermodynamic property Estimated Solvation Free energies (DGSolv)

These features were derived from all drug molecules in the dataset and used as input features for machine learning models.

Additionally, logP was included as an intermolecular property to quantify the hydrophobicity/hydrophilicity of a substance in water. This parameter provides a comprehensive measure of solute polarity, influencing its solubility behavior.

SASA was calculated from the average geometrical structure of the output trajectory using the Connolly algorithm, quantifying the molecular surface area exposed to the solvent. A higher SASA typically correlates with enhanced solubility due to increased molecular exposure to solvent interactions. Additionally, SASA is a valuable metric, alongside other molecular properties, for accurately assessing lipophilicity.

The solvation free energy was estimated by evaluating the solvation energy contribution of each atom per unit area of the solvent-accessible surface. The solvent orientation around the solute, defined as the cosine of the angle between the solvent dipole and the drug’s center of mass as a reference vector. Furthermore, the AvgShell metric, representing the average number of solvent molecules in the first solvation shell (0 – 0.5 nm), is a critical indicator of aqueous solubility. A higher AvgShell value suggests improved solvation and greater solubility. RMSD quantities a drug molecule’s structural stability and flexibility throughout the simulation. Lower RMSD values indicate a stable molecular conformation associated with well-defined and consistent solvent interactions.

The average number of hydrogen bonds is closely related to a molecule’s solubility, hydrophilicity, and hydrophobicity. Hydrophilic molecules, typically bearing polar groups, form hydrogen bonds that enable strong interaction with water molecules. The average number of hydrogen bonds, alongside the Coulombic and Lennard–Jones energies, quantifies the strength of intermolecular interactions.

It is worth mentioning that MD solubility simulations often rely on simplifying assumptions to make the problem computationally tractable while focusing on explicit aspects of molecular behavior. A prevalent assumption is the infinite dilution condition, wherein solute concentration is sufficiently low that solute–solute interactions are negligible compared to solute–solvent interactions. This condition enables a focused analysis of a single solute molecule’s interactions with the solvent, minimizing the complexity of solute aggregation or clustering. Consequently, Coulombic and Lennard–Jones energies were computed for the system’s solute–solvent and solvent–solvent interactions. This study aims to evaluate the significance of these properties through ML algorithms to ascertain their influence on aqueous solubility and improve its prediction.

Machine learning algorithms

A well-structured machine learning project involves several key components: data preprocessing, model training, hyperparameter tuning, and model evaluation. Data cleaning and partitioning into training and test sets are performed during preprocessing to ensure data quality and enable robust model validation. Subsequently, models are trained to select the appropriate algorithm, fitting it to the training data, thereby ensuring generalization to unseen data. Hyperparameter tuning is conducted to optimize the hyperparameters of selected models, enhancing their predictive capability. Finally, model performance is quantified using appropriate metrics to assess accuracy and reliability. This organized framework establishes a rigorous foundation for developing robust and reliable machine learning models.

Data preprocessing

The phase commences with identifying and eliminating outliers to ensure the dependability and precision of statistical models. Four methods were employed for outlier detection: Interquartile Range (IQR), Z Score, Isolation Forest, and Local Outlier Factor (LOF). Following outlier elimination, the dataset was partitioned into training and test sets with an 80/20 split ratio implemented via the Stratified splitting technique from the Verstack package (version 3.9.2). This method ensured the proportional distribution of the continuous target variable, logS, in both datasets. To minimize biases due to varying feature scales, the StandardScaler method was applied to scale each feature to a mean of zero and a standard deviation of one.

Models training

Ensemble methods have emerged as a prominent approach for enhancing predictive performance, leveraging the concept of the wisdom of crowds. This approach assumes that aggregating predictions from diverse models yields superior results compared to relying solely on a single model’s prediction. Ensemble methods categorized by their training strategies are broadly classified into two main types: Parallel and Sequential ensembles.

In Parallel ensemble methods, base learners are trained independently in parallel, categorized into sub-grouped: Homogeneous and Heterogeneous Parallel ensembles, distinguished by the type of base learners and training algorithms employed. Homogeneous ensembles utilize identical base learners, whereas heterogeneous ensembles incorporate diverse learners. Sequential ensemble methods take advantage of dependencies among base learners during the training process to enhance predictive performance. Each subsequent base learner is trained to mitigate errors committed by previously trained learners, thereby improving overall model accuracy. Sequential ensemble methods can be divided into two primary approaches: Adaptive Boosting and Gradient Boosting Ensemble. Adaptive Boosting trains consecutive base learners by dynamically adjusting their weights to focus on error correction from past iterations. However, Gradient Boosting further enhances the combination of gradient descent optimization with adaptive boosting to minimize the prediction error and yield more precise results.

This study investigates the application of sequential ensemble methods to predicting aqueous solubility by employing two homogeneous parallel ensemble algorithms, Random Forest (RF) and Extremely Randomized Trees (ExtraTrees), alongside two sequential gradient boosting algorithms: Gradient Boosting Regression (GBRT) and XGBoost. Bagging or Bootstrap aggregation represents the foundational homogeneous parallel ensemble technique. It utilizes a consistent learning algorithm and dataset, training base estimators on multiple bootstrap replicates of the dataset. Ensemble diversity is achieved through bootstrap sampling, also known as sampling with replacement, which introduces variability among base learners. Predictions from ensemble models are aggregated to produce a final output via two primary methods: majority voting or model averaging.

The Random Forest (RF) algorithm, pioneered by Leo Breiman and Adele Cutler⁴², represents a cornerstone of ensemble learning, renowned for its robustness and reliability across various applications, including drug discovery^43,44. RF constructs a collection of randomized decision trees as base learners through bagging and integrates their outputs via a collective voting process. Unlike conventional decision tree algorithms, which evaluate all features to determine the optimal split point for each decision node, Randomized Decision Trees (RDTs) employ a modified tree learning approach. RDTs introduce a random subset selection of features at each step, effectively lowering the inherent overfitting risk in isolated decision trees and yielding precise and consistent predictions. This approach reduces bias, while variance can be controlled by optimizing the hyperparameters of weak learners. Extremely randomized trees (ERT)⁴⁵ is a bagging-like ensemble method that constructs an ensemble of extremely randomized decision trees. ERT differs from other tree-based ensemble methods in two key aspects: it employs random selection of features and thresholds, increasing tree variability, and uses the entire dataset to train each decision tree rather than bootstrap replicas. This combination of Randomization and ensemble averaging reduces variance more robustly than other methods. Compared to parallel ensembles, sequential ensembles intend to combine multiple weak learners to form a single robust learner. A critical component of machine learning frameworks, the loss function facilitates the identification of the best-fitting model within the training set. It can treat all training set equally or prioritize specific instances by assigning them greater weights. Gradient boosting⁴⁶ a sequential ensemble method that aims trains a series of base estimators, typically decision trees by utilizing the gradient descent concept to minimize the loss value at each iteration. This process involves learning effectively from the residual errors of preceding models, repeating for a predetermined number of iterations or until predictive accuracy improvements diminish. The final prediction aggregates the contributions of all decision trees, adjusted by a learning rate known as shrinkage so that it until predictive accuracy improvements diminish. Newton boosting, another boosting approach, combines elements of AdaBoost and Gradient Boosting. Adaboost highlights misclassified examples by assigning them more wights, ensuring that subsequent base estimators prioritize these instances during the learning process. Conversely, Gradient boosting identifies misclassified examples through the gradient of the loss function, termed residuals. Newton boosting combines these strategies by employing weighted gradients to detect misclassified examples. eXtreme Gradient Boosting (XGBoost)⁴⁷, an open-source framework for tree-based Newton boosting, enhances this approach through regularized loss functions, efficient tree construction, and parallelizable implementation, making it highly effective for predictive modeling tasks.

Hyperparameter tuning

The performance and complexity of machine learning models depend highly on selecting hyperparameters in different model architectures. Identifying optimal hyperparameters is often challenging and necessitates systematically exploring of diverse configurations. A conventional approach to hyperparameter optimization is grid search, which performs an exhaustive search across a predefined subset of the hyperparameter space for a given estimator. This method systematically examines every possible combination of parameters and selects the most effective set based on a performance score. However, as each hyperparameter adds a new dimension to the search space, grid search can become computationally intensive, resembling brute-force approach with significant time requirements. Cross-validation (CV)⁴⁸ method is employed to avoid overfitting, typically through the GridSearchCV class in the Scikit-learn^49,50. CV is a resampling technique characterized by a single parameter, k, that determines the number of groups or folds, in which dataset is divided. The training data is randomly partitioned into k unique, equally sized, independent folds. The model is trained on k-1 folds and assessed on the remaining fold, with the process repeated k times to ensure each fold serves as the validation set once. Finally, the optimum hyperparameter set is determined by averaging the k performance scores, providing a robust estimate of model performance.

Evaluation metrics

The performance of the regression models was commonly assessed using two metrics: Root Mean Square Error (RMSE) and the coefficient of determination, (R²) (Eq. 1–2).

1

2

In Eq. 1–2n is the number of samples, is the ith real value and is the ith predicted value.

Feature selection

Irrelevant and redundant features can adversely affect ML model performance, compromising accuracy and computational resources. Therefore, feature selection (FS) plays a crucial role in identifying a compact subset of informative and robust features in the machine learning pipeline. This process not only reduces overfitting arising from better performance of models on training data rather than unseen data, but also improves model stability and declines its sensitivity to minor feature variations. FS methods are broadly classified into Filter, Wrapper, and Embedded techniques. Comprehension of the characteristics of different FS algorithms is beneficial for choosing the right one based on the dataset and modeling objectives. These characteristics encompass a range of factors, such as the type of analysis (univariate/multivariate), the learning paradigm (supervised/unsupervised/semi-supervised), and feature space search strategy (e.g., random, sequential, exponential, or greedy). Multivariate FS methods account for interdependencies and correlations among features, whereas univariate methods evaluate features independently, disregarding their relationships. Notably, not all FS methods belong to the univariate category⁵¹. Supervised Learning Techniques exert effort to use a combination of features with labeled target datasets to assess feature relevance, and all FS categories incorporate supervised learning within their evaluation metrics. Moreover, feature space exploration is accomplished sequentially, iteratively assessing feature contributions to the target variable. Filter-based FS methods examine feature subsets through statistical measures (i.e., correlation, consistency, information gain, or distance) without applying learning models. A fundamental aspect of filter methods is features’ ranking, which assigns each feature a relevance score. Features are retained or discarded based on a predefined threshold, informed by various ranking techniques. Spearman’s rank correlation coefficient, a statistical measure, is used to quantify a monotonic relationship between variables and targets, considering whether the relationship between two variables is linear, unlike Pearson’s correlation, which is limited to linear relationships.

Spearman’s rank is calculated per (Eq. 3) where ρ is the Spearman’s rank correlation coefficient, d_i is the difference between the ranks of the i-th pair of variables, and n is the number of observations. It ranges from -1 to 1, enabling straightforward interpretation of variable relationships. Values between 0 and 1 indicate a positive correlation, while values between -1 and 0 indicate an inverse correlation.

3

Wrapper-based FS techniques adopt a black-box approach, utilizing the performance output of classifiers, typically accuracy, as the evaluation criterion. These methods identify relevant features by assessing the improvement in model performance for selected feature subsets, as determined by machine learning algorithms. This study employs three wrapper-based approaches, namely Sequential Forward Selection (SFS)⁵², Sequential Backward Selection (SBS), and Recursive Feature Elimination with Cross Validation (RFECV). These methods are implemented using a range of classifiers, including: RF, XGBoost, Gradient Boosting, Extra Trees, Light Gradient Boosting, CatBoost, and AdaBoost. SFS procedure begins from an empty feature set and incrementally constructs a feature subset by evaluating the performance metrics derived from ML algorithms, continuing until a predefined number of remaining features. Conversely, SBS commences with the complete feature set and iteratively eliminates features based on poor performance scores, as determined by evaluation methods. RFE follows a logic similar to SBS, relying on feature ranking. It involves an iterative process of fitting a model using all features, and a relevance score is computed for each feature. Low-ranking features are subsequently eliminated, and the model is refitted with the updated feature subset until the optimal subset is identified. An important measure for selecting similarity between features for both forward–backward selection algorithms is the Symmetrical Uncertainty between features X and Y, as defined by (Eq. 4)⁵¹. Here, H(X) and H(Y) are the entropies of features X and Y, respectively, and H(X∣Y) is the conditional entropy of X given Y. This formulation, incorporating joint entropy H(X,Y) implicitly through the mutual information, guides the feature selection process to prioritize highly informative features.

4

Compared to Filter-based techniques, which are straightforward to implement and require reasonable computational resources, wrapper-based techniques suggest superior learning capabilities and accuracy. Through filter methods tend to be less accurate, wrapper methods are complex and costly. Embedded techniques seek to merge the strengths of both approaches, integrating search strategies with subset assessment through ML models. This study employs two embedded techniques: Importance-based selection and Shapley Additive exPlanation (SHAP)-value-based selection⁵³. Importance-based selection, increasingly utilized with Random Forest (RF), leverages RF’s inherent capacity to handle complex, high-dimensional datasets and capture feature interactions. Feature importance in RF is performed by measuring the impurity reduction attributable to each feature, with greater impurity reductions indicating higher feature significance. Features are then ranked by importance, and the most relevant are selected.

In contrast, SHAP, introduced by Lundberg and Lee⁵⁴, draws on game theory principles. This process involves two stages: first, training ML models using the entire feature set, and second, computing SHAP values for each feature. These values are ranked in descending order to identify and select the most impactful features⁵⁵.

Results and discussions

This study used experimental aqueous solubility of 211 drugs and related complex structures sourced from Huuskonen et al.³² (Table S4). MD simulations on this dataset yielded ten MD-derived properties that directly or indirectly influence solubility. The partition coefficient (logP), a critical property, was also collected from the literature. To ensure data quality and precision, the dataset was reduced from 211 to 199 due to the absence of reliable logP values for certain compounds (reverse transcriptase inhibitors, RTIs). During preprocessing, outlier detection was performed through statistical and ML techniques, excluding six compounds exhibiting extreme deviations from the mean. Subsequently, all properties were standardized to have a mean of zero and a standard deviation of one. The dataset was divided into a training set of 160 compounds for model development and a test set of 39 compounds to evaluate the predictive capability of models. Random selection ensured the diversity of drugs in both sets. A systematic feature selection strategy, combining statistical analysis (correlation assessment) and machine learning techniques (wrapper and embedded methods), was employed to determine the most effective features for solubility prediction. This approach ensured that the selected features were applicable to solubility prediction and significantly contributed to predictive performance. Four Ensemble ML models were trained on the dataset to evaluate the efficacy of seven selected features. The optimal parameters for each model were acquired through hyperparameter optimization, improving predictive accuracy. The resulting models demonstrated the ability to predict aqueous solubility with a reasonable degree of accuracy, explaining the effectiveness of the proposed feature selection and modeling approach.

Dataset analysis

Interpreting statistical descriptions of features is necessary for robust statistical analysis and ML applications. The normal distribution, called the Gaussian distribution, is a fundamental probability distribution widely employed in these domains. Its mathematical properties make computational efficiency and enhance the performance of algorithms, particularly those reliant on linear models. Data normalization or standardization, a standard preprocessing step, often assumes or produces a normal distribution to align with these algorithmic assumptions. Skewness quantifies the distortion from a symmetric distribution. Normal distribution, where the median, mean, and mode coincide, results in zero skewness and a symmetric distribution. A distribution is positively skewed if it shifts toward higher values (right-tailed) or negatively skewed if it shifts toward lower values (left-tailed). Before model training, the statistical description of features, including mean and median of features, was analyzed and reported in Table S1. This analysis provides insights into the distribution and range of each feature, informing subsequent modeling decisions.

Pairwise plots and Kernel Density Estimation (KDE) plots were used to visualize the dataset’s distribution. Pairwise plots, presented in Figure S1, are comprised of diagonal elements that depict the distribution shape of individual features and off-diagonal elements that illustrate the relationships between feature pairs, enabling a comprehensive assessment of feature interactions and distributions. KDE plots provide a visualization of the probability density of continuous variables. As depicted in Fig. 2, most features in the dataset are typically distributed, while others such as Hbond, Coulombic, and RMSD, which display skewed distributions. This skewness highlights significant difference across compounds for these specific properties, deviating from the symmetry of a normal distribution and suggesting relative uniformity in other features. Moreover, bimodality occurs in features like LJ_t and SASA, indicating the presence of two distinct data subgroups. The bimodality is likely attributable to compound subgroups within the dataset, characterized by differing molecular properties.

Fig. 2.

The collection of KDE plots, visualizes the distribution of diverse physicochemical properties, each exhibiting an approximately normal distribution with varying peaks and spreads across the x-axis.

Spearman’s rank correlation coefficient (ρ) was computed to assess correlation in two contexts: first, among all features to assess inter-feature correlations, and second, between each feature and the target variable to determine predictive relevance. A considerable absolute correlation coefficient value between two features suggests redundancy, leads to multicollinearity issues in the model, necessitating dropping or combining one of the correlated features. Features strongly correlated with the target variable are deemed highly relevant, indicating their substantial impact on the accurate prediction of the target. Figure 3 presents a heatmap of Spearman’s rank correlation coefficient, highlighting strong relationships among features. Notably, a strong negative correlation is observed between LJ_t and SASA (ρ = − 0.95) and between Coulombic and Hbond (ρ = − 0.92). These high correlations suggest potential redundancy, warranting consideration of either retaining only one feature from each pair or combining them. However, no features were excluded at this stage to thoroughly investigate the influence of all properties on aqueous solubility prediction. Beyond statistical insights, the heatmap elucidates a meaningful negative correlation between logS (aqueous solubility) and logP (partition coefficient), consistent with the principle that more hydrophobic compounds, characterized by higher logP values, exhibit reduced water solubility.

Fig. 3.

The Heatmap, displays Spearman’s correlation coefficients between solubility and various features in the dataset. The color gradient, ranging from blue (− 1.00) to red (1.00), indicates the strength and direction of correlations, providing insights into the relationships influencing solubility.

Outliers detection

Statistical analysis, as depicted in the box plots in Figure S2, revealed the presence of outliers in the dataset. Several detection techniques were exploited to identify outliers exactly, of which the Z score method was selected based on its consistency with results from other techniques. Consequently, identified outliers were removed from the dataset to ensure data integrity.

Figure 4 illustrates six drugs statistically identified as outliers and subsequently removed from the dataset. These compounds exhibit properties markedly distinct from the majority of the dataset. For instance, Triamcinolone displays elevated Coulombic and hydrogen bond values in MD simulations, chemically explainable by its abundance of polar functional groups. These groups contribute to a more uneven charge distribution and enhanced hydrogen bonding potential than other steroid drugs. Additionally, the Triamcinolone’s specific conformation and molecular interactions might further enhance these effects.

Fig. 4.

This figure, presents the chemical structures of drugs detected as outliers, showcasing the structural diversity among outliers in the dataset. The drugs depicted are: (a) Triamcinolone, (b) Prostaglandin_E2, (c) Thiopropazate, (d) Cefamandole, (e) 5-t-Butyl-5-_3-methylbut-2-enyl_barbiturate, (f) 5_5-Diphenylbarbituric_acid.

Feature selection

Two prominent FS approaches, Wrapper and Embedded, were utilized to identify the most influential features for predicting aqueous solubility. The Wrapper approach, employed three methods: SFS, SBS, and RFECV. These methods were trained using four ensemble methods (RF, Extra trees, GBR, and XGBoost RF) to identify optimal feature subsets Regarding the capability of SFS and SBS methods to predefine the number of anticipated features in advance, various subset sizes (ranging from 4 to 8 features) were evaluated across the four algorithms to determine both the optimal number and the most relevant features.

Figure 5 illustrates the CV R² scores for each method. The feature subsets yielding the highest CV R² scores (e.g., Extra Trees in SFS-4, RF in SFS-5) were identified as the most impactful features. Finally, each selected subset was used to train ML models, with results illustrated in Fig. 6. The similarity in features across selected subsets resulted in comparable R² scores in Fig. 6, indicating robust agreement among the feature subsets and the various ML methods between the selected features in each subgroup and different ML methods. The selected feature subsets are detailed in Table S3.

Fig. 5.

This bar chart, demonstrates cross validation R² score for four ML algorithms: RF, ExtraTrees, GBR and XGBoost, trained using various FS methods of wrapper approach, including SFS, SBS, and RFECV. For SBS and SFS methods, the performance is evaluated with different anticipated feature numbers ranging from 4 to 7, revealing the optimal feature sets that maximize predictive accuracy.

Fig. 6.

This bar chart, presents the test set R² score of four ML algorithms: RF, ExtraTrees, GBR and XGBoost, evaluated on selected feature subsets derived from maximum cross validation scores identified in the wrapper approach. The optimal feature sets, determined for each algorithm, are tested to assess their predictive performance in test set, highlighting the effectiveness of the selected features in enhancing model accuracy.

Through the applied FS methodology, a consistent set of features (logP, SASA, Coulombic_t, AvgShell, and DGSolv) emerged across all evaluated subsets, establishing them as key predictors of aqueous solubility.

To further prioritize these features, an embedded FS approach, including SHAP, and Random Forest (RF) feature importance, was employed.

As shown in Figs. 7 and 8, SHAP and RF feature importance analyses highlight the most influential features for solubility prediction. The SHAP plot compares the influence of each feature on the aqueous solubility output based on their SHAP values. It effectively detects expressive features and measures the extent of their contributions to the model. In this visualization, feature values are represented by a color gradient from blue (low) to red (high). In contrast, the SHAP value indicates the direction and magnitude of the impact, with positive values denoting a favorable effect on solubility and negative values indicating an adverse effect. The results from both SHAP and RF feature importance analyses converge on a core set of features, SASA, logP, Coulombic_t, LJ, DGSolv, and RMSD, as the most significant contributors to accurate solubility prediction, underscoring their critical role in the predictive modeling process.

Fig. 7.

This SHAP plot, illustrates the impact of individual features on the solubility output of a RF model. The x-axis represents the SHAP values indicating the magnitude and direction of each feature’s contribution, with a color gradient from blue (low feature value) to red (high feature value).

Fig. 8.

This chart displays the feature importance ranking for RF model, showing the relative contributions of features in predicting solubility. SASA and logP emerge as the most influential features with importance scores.

Analysis of wrapper and embedded approaches revealed four features (logP, SASA, Coulombic_t, DGSolv) that are consistently prominent across both methodologies. Additionally, three features (RMSD, AvgShell, and LJ) were selected from both approaches due to their importance in solubility. Consequently, these seven features were adopted for training ML models.

To further understand the importance of extracted features, two drug classes within the dataset (steroids and barbiturates) with an adequate sample size were analyzed individually using the Random Forest (RF) algorithm, as reported in Figure S3. The results confirmed that the same seven features were the most influential, although their relative rankings varied slightly between the drug classes. These variations are likely attributable to differences in the structural and thermodynamic properties of the compounds. The findings suggest that the chemical characteristics of the drugs modulate the importance of the features in the model drugs. Because of the diversity of drug categories in the dataset, ensuring a balanced representation for every class is crucial to thoroughly investigating the relationship between feature weights and the structural and thermodynamic properties of the compounds.

ML model training

To assess the predictive performance of the seven selected features for estimating aqueous solubility, four machine learning models (Random Forest, Extra Trees, XGBoost, and Gradient Boosting) were trained using these features. Hyperparameter optimization was conducted using a grid search algorithm to systematically explore all possible hyperparameter combinations, coupled with tenfold cross-validation to identify the optimal hyperparameter set for each model. Table S2 details the selected hyperparameters, and Table 1 presents the models’ performance metrics.

Table 1.

Models’ evaluation for aqueous solubility prediction.

ML models	Training set		Test set
	R2	RMSE	R2	RMSE
Random forest	95.19	0.318	83.33	0.613
Extra trees	95.24	0.316	84.06	0.600
XGBoost	94.60	0.337	84.48	0.592
Gradient boosting	96.33	0.277	87.20	0.537

All four models demonstrated robust predictive capabilities for aqueous solubility, achieving satisfactory R² values and RMSE. The best-performing model on the test set, Gradient Boosting, attained an R² of 0.87 and an RMSE of 0.537.

For comparison, Huuskonen et al.³² reported an R² of 0.86 and an RMSE of 0.53, using an artificial neural network (ANN) model with 23 structural descriptors on the same dataset of 211 drugs. Tayyebi et al.²⁴ investigated two distinct cheminformatics approaches (descriptor-based vs. fingerprint-based), focusing on static and structural features. They utilized 177 physicochemical descriptors and 2,048-bit ECFP4 fingerprints, reduced to 631 features via LASSO. The study employed Random Forest (RF) and Multiple Linear Regression (MLR), with RF achieving superior performance (R² = 0.88, RMSE = 0.64 for descriptors) compared to MLR. Ghanavati et al.⁵⁶ utilized three molecular representations for aqueous solubility prediction: Electrostatic Potential (ESP) maps derived from DFT calculations, molecular graphs using DeepChem, and tabular features combining ESP-derived properties with Mordred descriptors. They developed three ML models (EdgeConv, Graph Convolutional Network (GCN), and XGBoost) along with an ensemble model, achieving the best performance with the ensemble on the Solubility Challenge 2019 dataset (RMSE 0.865, R² ~ 0.92–0.93 across datasets). While many QSPR models rely on large sets of static structural descriptors or molecular fingerprints to achieve high accuracy, our work demonstrates that a compact and more physically interpretable set of just seven features, primarily derived from molecular dynamics, can yield comparable performance. The primary novelty of our study lies in validating the efficacy of features that describe the dynamic interactions between a solute and solvent (e.g., SASA, RMSD, interaction energies). This suggests that focusing on the dynamic behavior of molecules in solution provides a powerful and efficient alternative to high-dimensional, abstract descriptor sets, offering a clearer path toward understanding the mechanistic drivers of solubility.

Error analysis

Residual plots were generated to observe regression model performance by plotting test set residuals as the dependent variable against model predictions as the independent variables. These plots facilitate the identification of potential issues such as heteroscedasticity, non-linear relationships, or the presence of outliers in the dataset. Residual values should be randomly distributed around the horizontal zero line, indicating a well-fitted model. A cone-shaped pattern in the residual plot suggests data engineering or an alternative model, while a U-shaped pattern indicates that the model fails to capture non-linear relationships within the data.

Residual plots in Fig. 9 show a random distribution of values around the zero line, expressing the absence of systematic patterns or biases in the model predictions. Scatter plots in Fig. 9 illustrate the correlation between the predicted logS values derived from the ML models and experimentally measured values from the literature. In the test set, only two compounds (Cyclobutane-1,5 Spirobarbiturate, 6-hydroxypteridine) exhibited considerable deviations − 1.11 and − 1.01 log units, respectively. The Gradient Boosting Regression model demonstrated robust predictive performance across subgroups, particularly for Barbiturates and Xanthine, including their related compounds, with summed residuals of 0.013 and 0.487 log units, in the given order, demonstrating a prominent level of accuracy.

Fig. 9.

Performance evaluation of regression models: predicted vs. actual values and residual plots for random forest, extra trees, XGBoost, and Gradient Boosting Regressors.

Despite these strengths, limitations of the current model are recognized. Although the dataset was adequate for evaluating the influence of molecular dynamics (MD)-derived properties on aqueous solubility through machine learning, model generalizability could be improved by incorporating a larger and more diverse dataset, encompassing compounds with a broader range of aqueous solubilities, structural variations, and functional group characteristics. Furthermore, this study utilized MD-derived properties alongside the established experimental property, logP, for comparative analysis. The findings confirm that MD-derived properties exert a comparable influence on solubility prediction, validating their relevance in predictive modeling.

Conclusion

This study highlights the potential of combining molecular dynamics simulations with machine learning techniques to improve the accuracy and efficiency of aqueous solubility predictions. Practically, our MD-ML approach can be integrated into early-stage drug discovery pipelines as a screening tool to predict aqueous solubility for novel compounds, reducing reliance on resource-intensive experimental assays. By prioritizing candidates with optimal solubility profiles, this method can streamline compound selection, minimize synthesis costs, and accelerate development timelines. The experimental dataset of 211 pharmaceutical and related compounds’ aqueous solubility was gathered from Huuskonen et al.³² study. The comparison between MD-derived properties and traditional descriptors was conducted by selecting logP as a representative index of utilized descriptors. logP was selected due to its correlation with aqueous solubility in published studies. It allows us to compare the influence of MD-derived properties against this experimental parameter. logP values for this dataset were collected from literature sources; however, reliable values for 12 compounds could not be found. Since the performance of ML models intensely depends on data quality, these 12 drugs were excluded from the data to maintain the completeness and reliability of the dataset. The MD-derived properties and logP values of 199 drug molecules from different drug classes were constructed as features. Following careful data preprocessing, precise feature selection methods, consisting of statistical analysis and ML techniques (i.e., Wrapper and Embedded), were applied to our dataset. The conclusion of different methodologies in FS indicates that seven features—logP, SASA, Coulombic_t, DGSolv, RMSD, AvgShell, and LJ– were the most predictive. Two subgroups with adequate samples were considered individually to examine the importance of features in each subgroup of drugs. It was observed that the ranking of these selected features could vary slightly depending on the structural and thermodynamic properties of the samples.

The result of FS methods shows that MD-derived properties’ influence on aqueous solubility is considerable compared to the experimental property (logP). Having these features, four machine learning models—Random Forest, Extra Trees, XGBoost, and Gradient Boosting—were trained to predict aqueous solubility. Residual plots showed randomly distributed values around the zero line, indicating a lack of systematic bias and confirming the robustness of the models. The best predictive algorithm was Gradient boosting with an R² of 0.87 and an RMSE of 0.537. The comparison of the present model with previous studies shows that the performance of the predictive model of this study, using simpler and more interpretable algorithms besides just seven features derived from MD simulations as inputs, is comparable to models that employed a large number of features (descriptors, fingerprints) or complex ML algorithms. To improve the generalizability of the models to a wide range of chemical entities, future projects will aim to expand the dataset and consider other used descriptors, such as molecular fingerprints or topological indices, which would lead us to discover comprehensive effects of structural and dynamic properties on aqueous solubility. Furthermore, advanced ML algorithms will extend the capability of handling larger datasets and succeed in complicated relationships.

Software

All ML algorithms were implemented in Python (v3.8). NumPy (v1.26.4), Pandas (v2.1.4), SHAP (v0.45.0), and Scikit-learn (v1.2.2) were employed for data processing, model building, and interpreting outputs, respectively.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

Z. S. carried out the computer calculations, wrote the manuscript, and developed the theory. S. M. H. and S. E. contributed to the final version of the manuscript, designed and directed the project, and contributed to interpreting the results.

Data availability

The data supporting this study’s findings are available in the Supplementary Information files. Upon reasonable request, the corresponding author may provide additional details or raw datasets.

Declarations

Competing interests

The authors declare no competing interests.