We hypothesize that the first ancestral "ante-cell" molecular structures, i.e., first RNAs and peptides which gradually transformed into real cells once the Earth had cooled sufficiently for organic molecules to appear there, have leaved traces in RNAs and genes of present cells. We propose a circular RNA which could have been one of these ancestral structures whose vestigial pentameric subsequences would mark the evolution from this key moment when the ante-cells have begun to join the living organisms. In particular, we propose that in present RNAs (ribosomal or messenger) playing an important role in the metabolism of current cells, we look for traces of the proposed primitive structure in the form of pentamers (or longer fragments) belonging to their nucleotide sequence. The result obtained can be summarized in the existence of a gradient of occurrence of such pentamers, with a high frequency for the most vital functions (protein synthesis, nucleic synthesis, cell respiration, etc.). This gradient is also visible between organisms, from the oldest (Archaea) to the most recent (Eukaryotes) in evolution of species.