With the current explosion of genomic data, there is a greater need to draw inference on phenotypic information based on DNA sequence alone. We considered complete genomes from 35 diverse eukaryotic lineages, and discovered sets of proteins predictive of trophic mode, including a set of 485 proteins that are enriched among phagocytotic eukaryotes (organisms that internalize large particles). Our model is also predictive of other aspects of trophic mode, including photosynthesis and the ability to synthesize a set of organic compounds needed for growth (prototrophy for those molecules). We applied our model to the Asgard archaea, a group of uncultured microorganisms that show close affinities to eukaryotes, to test whether the organisms are capable of phagocytosis, a phenotypic trait often considered a prerequisite for mitochondrial acquisition. Our analyses suggest that members of the Asgard archaea - despite having some eukaryote-specific protein families not found in other prokaryotes - do not use phagocytosis. Moreover, our data suggest that the process of phagocytosis arose from a combination of both archaeal and bacterial components, but also required additional eukaryote-specific innovations.