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Cytotoxic lymphocytes are key elements of the immune system that are primarily responsible for targeting cells infected with intracellular pathogens, or cells that have become malignantly transformed. Target cells are killed mainly via lymphocyte exocytosis of specialized lysosomes containing perforin, a pore-forming protein, and granzymes, which are proteases that induce apoptosis. Due to its central role in lymphocyte biology, as well as its implication in a host of pathologies from cancer to autoimmunity, the granzyme-perforin pathway has been the subject of extensive investigation. Nevertheless, the details of exactly how granzyme and perforin cooperate to induce target-cell death remain controversial. To further investigate this system, we developed a biophysical model of the immunological synapse between a cytotoxic lymphocyte and a target cell using a spatial stochastic simulation algorithm. We used this model to calculate the spatiotemporal evolution of granzyme B and perforin from the time of their exocytosis to granzyme internalization by the target cell. We used a metric of granzyme internalization to delineate which biological processes were critical for successful target-cell lysis. We found that the high aspect ratio of the immunological synapse was insufficient in this regard, and that molecular crowding within the synapse is critical to preserve sufficient concentrations of perforin and granzyme for consistent pore formation and granzyme transfer to target cells. However, even when pore formation occurs in our model, a large amount of both granzyme and perforin still escape from the synapse. We argue that a tight seal between the cytotoxic lymphocyte and its target cell is not required to avoid bystander killing. Instead, we propose that the requirement for spatiotemporal colocalization of granzyme and perforin acts as an effective bimolecular filter to ensure target specificity.