Biological membranes are fundamental components of living organisms that play an undeniable role in their survival. Molecular dynamics (MD) serves as an essential computational tool for studying biomembranes on molecular and atomistic scales. The status quo of MD simulations of biomembranes studies a nanometer-sized membrane patch periodically extended under periodic boundary conditions (PBCs). In nature, membranes are usually composed of different lipids in their two layers (referred to as leaflets). This compositional asymmetry imposes a fixed ratio of lipid numbers between the two leaflets in a periodically constrained membrane, which needs to be set appropriately. The widely adopted methods of defining a leaflet lipid ratio suffer from the lack of control over the mechanical tension of each leaflet, which could significantly influence research findings. In this study, we investigate the role of membrane-building protocol and the resulting initial stress state on the interaction between small molecules and asymmetric membranes. We model the outer membrane of bacteria using two different building protocols and probe their interactions with the quinolone signal (PQS). Our results show that differential stress could shift the position of free energy minimum for the PQS molecule between the two leaflets of the asymmetric membrane. This work provides critical insights into the relationship between the initial per-leaflet tension and the spontaneous intercalation of PQS.