The OTE-24 method is a macromolecular sequence characterization technique based on the integration of the physicochemical properties of amino acids using a bigram approach. It involves computing matrices that represent the distances between amino acids according to properties such as hydrophobicity and hydrophilicity, and then extracting feature vectors by combining these matrices through correlation and concatenation methods. This approach captures both the local order of amino acids and their immediate contextual relationships within the sequence [8] .

The OTE-24 approach offers the advantage of generating normalized, fixed-dimensional feature vectors, facilitating their use in classification models. It effectively exploits local relationships between amino acids, which is crucial for identifying functional or structural motifs. Furthermore, the combination of multiple physicochemical properties enhances the richness of the extracted information, contributing to improved prediction accuracy for molecular interactions.

However, a notable limitation of this method is its limited capacity to model long-distance interactions between amino acids, which play a critical role in the three-dimensional structure and function of proteins. This restriction to the analysis of local relationships can reduce the model’s precision when predicting complex structural behaviors or interactions that require a more global overview of the sequence.

In macromolecular structures, it is common for amino acids that are distant within the sequence to interact directly within the three-dimensional structure. These interactions play a decisive role in the stability and function of biological macromolecules. However, the OTE-24 method, limited to successive amino acid pairs, is unable to capture these long-distance relationships. To address this limitation, it is necessary to introduce a mechanism capable of quantifying the correlations between the physicochemical property values of amino acids separated by a given distance. The proposed extension, named OTE-24LD, consists of enhancing the classical OTE-24 descriptor by integrating weighted correlations at various distances. The idea is to preserve the local information provided by OTE-24 while adding components that represent interactions between amino acids separated by a defined number of positions within the sequence. Taking these weighted correlations into account improves the model’s ability to capture the complexity of molecular structures and interactions at multiple scales.

Let a protein sequence of length $L$ be composed of amino acids denoted as $A_1$, $A_2$, $A_3$, $A_4$, $\cdots$, $A_L$. Each amino acid $A_k$ is associated with a numerical value for each considered physicochemical property, denoted as $C_{\left(k,p\right)}$, where $p$ represents the property index (in this study, $P=2$). The long-distance correlation between amino acids separated by $d$ positions is calculated as follows:

$V_d=\underset{k=1}{\overset{L-d}{{\displaystyle \sum }}}\left({\omega }_d\cdot \underset{p=1}{\overset{P}{{\displaystyle \sum }}}{C}_{A_k,p}{C}_{A_{k+d},p}\right)\text{with}1\le k\le L-1$(1)

where:

$d$ represents the distance between two amino acids, varying from 1 to $D_{\max }$,

${\omega }_d$ is the weight associated with the distance d, and

$P$ is the number of physicochemical properties considered.

This formulation makes it possible to quantify the interaction between all pairs of amino acids separated by $d$ positions for each of the selected properties.

To modulate the importance of interactions based on distance, we introduce a weighting factor ${\omega }_d$, which follows an exponential decay defined as:

${\omega }_d={\text{e}}^{-\alpha d}$(2)

where $\alpha$ is an adjustable parameter that controls the rate of decrease in the influence of interactions as the distance increases. This mechanism preserves the effect of nearby correlations while progressively reducing the influence of more distant relationships.

For each macromolecular sequence, the $V_d$ values are computed for all distances $d$ ranging from 1 to $D_{\max }$. The results are then concatenated to form a unique feature vector:

$V=\left[V_1{V}_2{V}_3\cdots V_{D_{\max }},V_{2,1},V_{2,2},V_{2,3}\cdots V_{2,20}\right]$(3)

This vector thus brings together all the weighted long-distance correlations between amino acids for each of the selected properties. In this study, we considered two properties: hydrophobicity and hydrophilicity.

For this OTE-24LD extension, two essential properties were selected. Hydrophobicity, a key property in the structuring of macromolecules and the formation of internal hydrophobic regions, and hydrophilicity, which influences the exposure of amino acids to solvents and interactions with the aqueous environment. These two properties were chosen because they are widely used in studies of macromolecular structural behavior.

In this section, a class of possible solutions is proposed and the insertion of these solutions in the main model is investigated to check where they will lead to. This is initiated by the statement of the following claim.

The study was conducted using data from the Human Protein Reference Database (HPRD), a reference resource for the functional and structural analysis of human proteins [13] . Developed through a collaboration between the Bioinformatics Institute of Bangalore (India) and the Pandey Laboratory at Johns Hopkins University (USA), HPRD compiles manually curated scientific annotations covering the majority of human macromolecules.

The latest version of this dataset includes over 36,500 unique macromolecular interactions involving 25,000 proteins and 6360 isoforms [14] [15] .

In this study, we maintained the same data proportions as the original OTE-24 method [8] . We performed preprocessing on the macromolecular sequences prior to feature extraction. First, incomplete and redundant sequences were filtered out to ensure the integrity and independence of the samples. Then, verification and homogenization of the macromolecular or protein alphabet al. allowed us to retain only sequences containing the 20 standard amino acids. Finally, the physicochemical properties of amino acids used in the descriptor calculations were normalized using the following transformation:

$C_{A,p}^{\text{norm}}=\frac{C_{A,p}-C_p^{\text{min}}}{C_p^{\text{max}}-C_p^{\text{min}}}$(4)

where $C_{A,p}$ is the raw value of a given property for amino acid A, and $C_p^{\min }$ and $C_p^{\max }$ are respectively the minimum and maximum values of this property across the 20 standard amino acids.

The parameters involved in the computation of long-range correlation descriptors were optimized based on preliminary experiments conducted on the HPRD dataset. A grid search procedure was employed to identify the most relevant combinations of values for the parameters $D_{\max }$ and $\alpha$, using a validation set extracted from the training data (80% of the HPRD dataset).

The parameter $D_{\max }$, which defines the maximum distance between two amino acids considered for computing correlations between physicochemical properties, was evaluated over the range 10 to 30 with a step size of 5. The highest average performance was achieved for $D_{\max }=20$, which was therefore selected for the final model.

Similarly, the exponential weighting factor $\alpha$, which controls the attenuation of interaction influence as a function of distance, was explored within the interval [0.01, 0.10], with an increment of 0.01. The optimal value identified was $\alpha =0.05$.

With respect to the physicochemical properties considered, two fundamental attributes were retained: hydrophobicity and hydrophilicity, due to their crucial role in protein−protein interaction mechanisms.

To ensure an unbiased performance assessment and minimize training-related biases, we adopted a stratified 5-fold cross-validation procedure in conjunction with the Random Forest algorithm. The HPRD dataset was randomly partitioned into five balanced subsets based on class labels (positive/negative). In each iteration, four subsets were used for training and the remaining one for testing. This process was repeated five times, with each subset serving once as the test set. Final performance metrics were computed as the average of the scores obtained across the five iterations.

To highlight the performance of the OTE-24LD method, it was compared to several classical descriptors from the literature, including AAC, DPC, CTD, PseAAC, and OTE-24. Recall that the AAC (Amino Acid Composition) model encodes the proportion of each amino acid within the sequence without order information [16] [17] . The DPC (Dipeptide Composition) model quantifies the frequency of successive amino acid pairs, thereby incorporating local order information [18] [19] . The CTD (Composition, Transition, Distribution) encodes sequences based on groupings of physicochemical properties and their distribution [20] [21] . The PseAAC (Pseudo Amino Acid Composition) enriches the AAC encoding by considering correlations between distant positions [22] [23] . Finally, the OTE-24 technique is a coding method based on weighted local interactions, considered here as the immediate reference method.

We applied these different methods on the same HPRD dataset and used the same cross-validation procedure, thus ensuring a fair and rigorous comparison of their performance.

In this section, we present the performance of the proposed OTE-24LD model, trained using the Random Forest algorithm, for the prediction of interactions between biological macromolecules. The performance was assessed using several metrics relevant to the study context. These metrics reflect different aspects of the model’s predictive behavior, particularly in the context of imbalanced classes. We used accuracy to provide a general measure of prediction correctness, while precision, recall, and specificity assess the model’s ability to correctly identify positive and negative interactions. The F1-score offers a meaningful balance between precision and recall, especially useful in imbalanced class scenarios. Balanced accuracy corrects for the impact of class imbalance, and the Matthews Correlation Coefficient (MCC) provides a robust evaluation even when classes are unequal. Additionally, the ROC and PR curves, along with their respective areas under the curve (AUC and AUPR), enable graphical and quantitative analysis of the model’s performance across various classification thresholds. The confusion matrix completes this analysis by offering a detailed visualization of prediction errors. These metrics were selected to ensure a rigorous and multidimensional evaluation of performance, tailored to the specific contextual characteristics of the problem studied. They are calculated as follows:

$\text{Accuracy}=\frac{\text{TP}+\text{TN}}{\text{TP}+\text{TN}+\text{FP}+\text{FN}}$(5)

$\text{Precision}=\frac{\text{TP}}{\text{TP}+\text{FP}}$(6)

$\text{Recall}=\frac{\text{TP}}{\text{TP}+\text{FN}}$(7)

$\text{specificity}=\frac{\text{TN}}{\text{TN}+\text{FP}}$(8)

$\text{F}1\text{-score}=2\times \frac{\text{Precision}\times \text{Sensibilite}}{\text{Precision}+\text{Sensibilite}}$(9)

$\text{Balanced Accuracy}=\frac{\text{Sensibilite}\times \text{Specificite}}{2}$(10)

$\text{MCC}=\frac{\left(\text{TP}\times \text{TN}\right)-\left(\text{FP}\times \text{FN}\right)}{\sqrt{\left(\text{TP}+\text{FP}\right)\left(\text{TP}+\text{FN}\right)\left(\text{TN}+\text{FP}\right)\left(\text{TN}+\text{FN}\right)}}$(11)

Together, these indicators allow for a comprehensive analysis of the model’s performance, both in terms of overall prediction accuracy and the differentiation between positive and negative classes. The following table summarizes the performances obtained across all test sets ( Table 1 ).

The model achieved an accuracy of 93.51%, indicating that a large proportion of the total predictions were correct. However, in contexts where class imbalance exists, accuracy alone can be misleading. For this reason, balanced accuracy was also computed, reaching 93.52%, reflecting a balanced ability to correctly predict both classes. The recall of 90.10% demonstrates the model’s ability to correctly identify positive cases, while the specificity of 96.94% highlights its capacity to avoid false positives. The F1-score, which synthesizes precision and recall, reached 93.30%, indicating a good trade-off between interaction detection and error minimization. The Matthews Correlation Coefficient (MCC), recognized for its robustness in imbalanced class scenarios, achieved 87.23%, further confirming the overall quality of the model.

To assess the effectiveness of our proposed approach, we conducted a comparative evaluation against several widely recognized models from the literature, including AAC, DPC, and PseAAC, each coupled with different classifiers. The original OTE-24 model, primarily designed to capture short-range Bigram interactions, was used as a baseline in our experiments. It achieved an accuracy of 86.61%, an F1-score of 87.33%, a ROC-AUC of 89.43%, and a Matthews Correlation Coefficient (MCC) of 86.19. In contrast, our enhanced method, OTE-24LD, which incorporates both physicochemical properties and long-range interaction features, demonstrates a 6.73% improvement in accuracy over OTE-24.

This performance gain is further supported by the results presented in the table below, which compares our approach to conventional methods that also consider long-distance interactions ( Table 2 ).

The OTE-24LD method outperforms classical approaches in terms of accuracy, F1-score, and AUC, confirming the value of integrating physicochemical properties and long-distance interactions.

The OTE-24LD (Long Distance) method stands out from existing approaches in the literature by explicitly integrating long-range correlations between residues. While conventional descriptors such as AAC and PseAAC limited to local frequencies or immediate neighborhoods struggle to exceed 70% accuracy, OTE-24LD achieves a global accuracy of 93.51%. This significant improvement highlights the importance of capturing extended sequential dependencies in the prediction of macromolecular interactions.

This enhancement is also reflected in the F1-score, which increases from 82.50% for DPC (used as the reference classifier) to 93.30% with OTE-24LD. Such a gain demonstrates a better balance between recall and precision, made possible by our adaptive weighting scheme. This mechanism preserves the influence of distant residues without introducing redundancy or diluting relevant local information. The AUC reaches 97.99%, a value close to the theoretical optimum, indicating an excellent trade-off between sensitivity and specificity. Likewise, the Matthews Correlation Coefficient (MCC), a metric on which traditional methods often underperform in this context, reaches 0.8723, reflecting a well-calibrated and reliable model capable of limiting both false positives and false negatives.

Beyond these quantitative results, the ability to capture spatially distant but functionally coupled residues enables more accurate prediction of interaction interfaces, better modeling of complex stability, and identification of key regions involved in molecular recognition. This level of precision contributes to a deeper understanding of interaction mechanisms and supports the rational design of therapeutic biomolecules by reducing dependence on costly and time-consuming in vitro experiments.

Nevertheless, the OTE-24LD approach presents certain limitations that should be acknowledged. It relies solely on information derived from the primary sequence and does not incorporate structural or evolutionary features, which could further refine predictive performance. Additionally, computing long-distance correlation descriptors incurs a non-negligible computational cost. The simulations conducted in this study were performed on a high-performance server (25.1 GHz CPU, 128 GB RAM, 830 GB SSD storage). Although resource consumption remained within reasonable limits, this requirement may constitute a constraint for large-scale or real-time applications without further optimization.

By reducing reliance on experimental validation, OTE-24LD has the potential to accelerate the development and refinement of therapeutic candidates. This approach offers a robust, relevant, and generalizable solution for leveraging long-range sequence information in the prediction of macromolecular interactions. Through its performance, interpretability, and practical scope, it represents a promising advancement for research in structural and functional bioinformatics.