A drug is a type of natural or synthetic chemical that is used to prevent, treat, or diagnose disease [1]. It can be able to alter the function of a biological system or target from the molecular to the cellular level. Drug discovery helps to determine new therapeutic candidates by using different computational, experimental, and clinical models. The integrated approaches led to the identification of novel drugs not only from plants but also from other chemical sources [2]. Although various therapeutic compounds originating from plant products are highly regarded, synthetic chemistry and biotechnology products account for the majority of medications in the current medical system [3]. The subject of drug development is exceedingly difficult and needs proper infrastructure and laboratory resources. Unfortunately, the traditional strategy of discovering new drug compounds is a time-consuming process that can take up to 10–15 years and can cost up to USD 2.558 billion to bring a therapeutic to market [4]. This is a multistage and complex process that begins with the identification of an appropriate drug target, followed by drug target validation, hit-to-lead identification, and lead molecule optimization, as well as preclinical and clinical research [5]. Despite the huge financial and time commitments required for medication development, clinical trial success is just 13%, with a high drug attrition rate [6].

A mathematical model is a powerful representation of a biological system that uses mathematical ideas and language to produce an accurate description of the system of principles [7]. The model helps in determining the operation process as well as anticipating certain influencing factors and enables the simulation of complex biological processes that generate hypotheses and suggest experiments [8]. The model also known as forecasting modeling is now frequently used to guide drug development at the industrial level. For example, simulation is the more direct approach that utilizes a mathematical model and predicts system behavior under given conditions [9]. Mathematical model-based biological complex system analysis has high productivity and low cost. The process generates novel lead compounds that undergo clinical trials and reach the market [10]. Most of the major obstacles that arose during the conventional drug design and discovery process may be overcome by employing mathematical models [11]. These models are now being utilized in in silico research to describe various pharmacological properties of potential medicinal drugs [12]. For example, the FDA’s Center for Drug Evaluation and Research (CDER) uses modeling and computer simulations at various phases of drug discovery [13].

Currently, CADD has proven to be a useful and powerful strategy in the manufacture of various medicines [14]. The approach has assisted in overcoming the drawbacks of a time-consuming and expensive procedure in drug research and development [15]. In the latest drug design process, the in silico approach is more important than before. CADD methods such as pharmacophore modeling, virtual screening, molecular docking, and dynamic simulation are frequently applied to identify, develop, and evaluate medicinal properties as well as comparable physiological activity of substances [16]. To quantify the binding efficiency and toxicity of a compound in the classical drug development process, massive in vitro and in vivo trials are required [17]. CADD techniques include a molecular docking methodology that can effectively categorize a large number of molecules with higher binding effectiveness [18]. The method can be used to identify the interaction between a ligand and a receptor at the atomic scale, which helps to identify the binding position of a molecular to a target protein and subsequently provides an idea about the biochemical process [19]. The technique also provides information regarding the target behavior and predicts how a protein (enzyme) interacts with small molecules (ligands) at the binding site of target proteins and facilitates the evaluation of the biological activity of a molecular candidate [20]. Additionally, the CADD approaches include different pharmacology properties analysis tools that can evaluate a compound’s pharmacokinetic (PK) parameters such as bioavailability, toxicity, and effectiveness within a short period. Furthermore, the CADD approaches also include molecular dynamics (MD) simulation techniques that can determine a ligand’s binding stability towards its receptor, which is more suitable and accurate [21]. 

2. Target Identification

The early stages of drug discovery probably start with target selection and later move to lead optimization. In the process of potential disease, target discovery is dependent on a variety of resources, involving academic studies, clinical investigations, and the business sector. The pharmaceutical industry, as well as numerous research organizations, use the designated target to locate molecules for developing authorized treatments [22]. Several preliminary stages are involved in this procedure. Throughout the process of target identification and validation, researchers search for chemicals to disrupt a particular biological path that is connected to a certain illness [23]. These compounds can be found in nature, identified through high-throughput screening of large compound libraries, or synthesized as analogs of other drugs that have been proven to be effective against a specific disease. The initial stages in target classification and identification are to determine the function of a possible therapeutic target (which may be a gene or protein) and its involvement in the illness [24]. The molecular processes addressed by the objective are characterized by the following target identification. A good target must always be productive, safe, suitable, and druggable, and it must fulfill clinical and financial requirements. Target identification may be divided into two types: the system biology approach and the molecular biology approach. The system biology approach is a technique that involves studying diseases in complete organisms and selecting targets based on data from clinical trials and in vivo animal research [25]. The molecular biology method, which is at the heart of today’s target identification efforts, aims to find “druggable” targets whose activity may be influenced by associations with molecules, proteins, and sometimes antibodies. Since the biological factors involved in human diseases are so complicated, the foremost essential issue in target identification is not only identifying, optimizing, and choosing trustworthy “druggable” targets, but also truly comprehending the cell membrane associations that identify disease patterns, developing predictive models, and building biological mechanisms for human diseases [26]. For example, G-protein-coupled receptors (GPCRs) and protein kinases are highly “druggable” targets that were identified throughout the molecular biology-based methods [27]. Network-based drug discovery, a field that utilizes information in drug–protein and protein–disease networks, may also be used to study target identification [28]. This strategy entails a highly collaborative scheme between databases and correlations across genomics, transcriptomics, proteomics, metabolomics, the study of the microbiome, and pharmacogenomics, and it is heavily reliant on the development of relevant mathematical, computational, and systems biology tools that connect pharmacological and genomic domains and create computational frameworks for drug target discovery [29]. Another recent network-based application was the combination of large-scale structural genomics and disease association studies to produce a three-dimensional human interactome, which resulted in the identification of candidate genes for previously unknown disease-to-gene associations with proposed molecular mechanisms.

3. Mathematical Models in Drug Design

Mathematical techniques for drug discovery have a high value because of their potential effect and low cost compared to preclinical studies [30]. The employment of mathematical models, as well as computer simulations, has several advantages. It can be very helpful for systematically determining the relevance of a specific target or pathway for the overall behavior of the system. First, the inconsistencies between the behavior forecasted by a mathematical model and the behavior observed in actual trials might point to missing components, in which the mathematical model allows for a briefer image of a biological mechanism to develop. Although it is not clear which compounds are absent from the system under review, the mathematical model research findings may be used to influence the construction of additional investigations to address the problem. In addition, mathematical models enable a systematic analysis of system fluctuations triggered by the delivery of drugs [31]. However, it is difficult to represent real-world systems such as biological systems in terms of mathematical relationships [32]. Figure 1 shows the process of the use of a mathematical model in the drug design process. Pharmacokinetic and pharmacodynamic analyses are the earliest and most widely used forms of mathematics in drug design. Pharmacokinetics is the study that describes how drug concentrations change over time, whereas pharmacodynamics explains how drug effects fluctuate with concentration. Pharmacokinetics depict a possible drug’s concentration in the appropriate organ compartments (e.g., circulating blood). Pharmacodynamic models relate this concentration to a biomarker that is thought to be linked with a disease state, often considering the modification of the pharmaceutical target [33]. Cancer research is a good example of how mathematical models are used in drug discovery. One of the most widely employed mathematical models in cancer treatment research is integrated into network-based medicine [34]. Network medicine is a discipline of medicine that explores molecular and physiological links with therapeutic implications. Infectious diseases, such as malaria, are another instance of a mathematical model application in drug innovation [35]. In this situation, mathematical models may be employed to evaluate the prospective drug’s capacity to destroy the parasite at a different phase of the disease. Compound pharmacokinetics and compound pharmacodynamics are used in such models. COVID-19, an infectious viral disease, is the most recent example of how mathematical models are employed in drug discovery [36].

4. Protein Structure Prediction

Proteins are vital molecules that are involved in a variety of biological activities. Protein structure prediction or modeling is critical since a protein’s activity is largely determined by its three-dimensional structure. Furthermore, a protein’s 3D structure is determined by its amino acid composition. Experiments using X-ray crystallography or NMR spectroscopy to resolve protein structure are time-consuming, expensive, and complex ^[37][42]. Consequently, theoretical knowledge of protein structure, dynamics, and folding has been used to construct a model from amino acid sequences due to the improvement of computer methods and computational tools. The approaches for predicting protein structure may be divided into three categories (Figure 2): (a) homology modeling; (b) threading; (c) ab initio methods (de novo). The most effective computer technique for protein structure prediction is homology modeling, which involves predicting an unknown structure using a similar known protein structure as a framework ^[38][43]. An ideal therapeutic simulation of a protein may be built by assigning a structure based on sequence alignment and then creating the model and minimizing energy. Despite homology modeling’s predictive potential and utility, some issues remain. Firstly, the amounts of target-template architectural conservation and alignment precision are key indicators of the model’s quality. If the identity of the template sequence is below 20%, around 50% of residues inside the layout are likely to be misaligned. Another concern includes that homology modeling systems should develop innovative ways to manage the expanding number of existing protein molecules. To date, different homology modeling tools has been developed and the most frequent use tools use for the modeling has been listed in Table 1. Threading a sequence throughout a fold involves a precise adjustment of the protein’s amino acid sequence with the folding motif’s corresponding amino acid residue residues. The main goal of this technique is to determine the most possible fold from a given sequence or to find appropriate sequences that might fold into a certain structure. Threading performance is characterized by the number of useable folds whose structures are determined precisely towards the atomic level ^[39][44]. Threading processes, which use approaches for aligning sequences with 3D shapes to determine the proper folding of a given sequence from a range of possibilities, were used to make the predictions. In absence of an experimentally solved structure of a similar/homologous protein, ab initio (de novo) protein structure prediction is a technique for evaluating the three-dimensional structure, when an experimentally solved structure of a similar/homologous protein is not present. The energy function guides the construction of protein structure in this strategy. The ab initio (from scratch) methodologies are based on first-principles physics and chemistry regulations, as well as the premise that a protein’s natural structure always remains at the lowest energy level ^[40][45]. However, the precision of ab initio modeling is poor, and performance is generally limited to tiny proteins (120 residues).