## 2 Main Aspects of PTML Models for MOOP  
In general terms, PTML models can be viewed as multi-objective quantitative structure-activity relationship models, which means they predict molecular behavior based on chemical structure and can be optimized for multiple outcomes, such as potency and pharmacokinetic properties [29]. Following this, we summarize the main aspects that make PTML modeling a suitable approach for MOOP applied to drug discovery (Figure 1):  
Figure 1: Main steps leading to the development and application of a PTML model for antibacterial discovery. * Within the MOOP paradigm, PTML models can be used to predict outcomes by simultaneously considering multiple objectives such as biological effects, targets, and assay protocols. * It is possible to physicochemically and structurally interpret PTML models. * The PTML modeling approach is highly adaptable, which enables its application to different chemistry realms, such as small organic molecules, peptides, and MCNPs. We will now focus on giving more details on the three aspects mentioned above. We want to start by stating that the key element of PTML modeling is the application of the Box-Jenkins approach [29]. Let us consider any experimental condition as a combination of three aspects, namely measures of endpoints/biological effects, targets, and assay protocols. Each of these three aspects has multiple categories/labels. For instance, the aspects known as \"targets\" will contain multiple categories/labels corresponding to the names of different targets (microbial strains, cell lines, laboratory animals, etc.). The same line of thought can be applied to the other two aspects. The power of the Box-Jenkins approach relies on fusing the chemical information (characterized utilizing molecular descriptors) of the molecules with the biological information provided by the different aspects (and their multiple labels) of the experimental condition [29]. As a result, new numeric variables known as multi-label descriptors (_MLDs_) will be created, integrating both chemical data related to the structure of the molecules and biological information on various biological endpoints, targets, and assay protocols [29]. Depending on its purpose, a PTML model can be further subdivided into multi-target models for quantitative structure-activity relationships (mt-QSAR), multi-condition QSAR (mtc-QSAR), or multi-tasking quantitative structure-biological effect relationships (mtk-QSBER) [29]. In the case of mt-QSAR models, they can predict a single activity endpoint by simultaneously considering multiple targets. In the case of mtc-QSAR models, they can predict multiple activity endpoints against different activity-related targets. Notice that multi-target drug discovery can be considered as a special case of MOOP where one or more activity/potency endpoints can be predicted by a PTML (mt-QSAR or mtc-QSAR) model against different biomolecular targets (usually proteins). Finally, mtk-QSBER models represent the most advanced evolution of PTML modeling and can predict multiple _in vitro_ and _in vivo_ activity, toxicity, and pharmacokinetic endpoints by considering multiple targets and many assay protocols. For the case of simplicity, we will use the unified term PTML to refer to all models discussed in this review. When applied to antibacterial discovery, PTML models can be employed to simultaneously predict antibacterial activity endpoints (e.g., minimum inhibitory concentration, minimum bactericidal concentration, etc.) across different targets (bacterial proteins or strains) and many assay protocols (i.e., MTT, resazurin, and others). Additionally, PTML models can incorporate toxicity and pharmacokinetic data for molecules, even when these differ from those with reported antibacterial activity outcomes. Currently, PTML models are interpretable. However, the interpretation of PTML models differs from other approaches, such as explainable artificial intelligence (XAI) [55; 56; 57; 58].