Accumulating data suggest that the eukaryotic cell originated from a merger of two prokaryotes, an archaeal host and a bacterial endosymbiont. However, since prokaryotes are unable to perform phagocytosis, the means by which the endosymbiont entered its host is an enigma. We suggest that a predatory or parasitic interaction between prokaryotes provides a reasonable explanation for this conundrum. According to the model presented here, the host in this interaction was an anaerobic archaeon with a periplasm-like space. The predator was a small (facultative) aerobic alpha-proteobacterium, which penetrated and replicated within the host periplasm, and later became the mitochondria. Plausible conditions under which this interaction took place and circumstances that may have led to the contemporary complex eukaryotic cell are discussed.