Constraints of a multireplicon chromosomal organization and of the necessity to maintain constant gene dosages demand that each origin of replication in a eukaryotic cell "fire" (initiate replication) only once per cell cycle. The central idea of this work is that a low probability of an extra ("illegitimate") round of DNA replication (called below "replicon misfiring") within any given chromosomal domain could be increased by certain substances of either intra- or extracellular origin. The term " "firone" is proposed for such a substance. It is shown that existence of firones could greatly speed up evolution of cellular systems under selection pressure, a developing tumor being one example of such a system. Experimentally testable predictions of the firone hypothesis are discussed.