You would know that eukaryotes evolved from prokaryotes. So, the interesting point of the question is whether sleep is something that occurs at the cellular le

You would know that eukaryotes evolved from prokaryotes. So, the interesting point of the question is whether sleep is something that occurs at the cellular level or the organism level. See the article linked to in the comment above, about cyanobacteria. > Cyanobacteria maintain their circadian rhythms even when isolated from the naturally occurring daily light-dark cycles of the sun, just as humans do. The researchers found that under conditions of moderate constant light, the cyanobacteria undergo cell division about once per day, and the divisions take place mostly at the midpoint of the 24-hour cycle.