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.One feature ofeukaryotic DNA replication is tight control ofthe number ofinitiation events per origin ofreplication. Apart from a few special cases (2-5), each origin of replication in a eukaryotic cell fires only once per cell cycle; repetitive firings at the same origin are prevented, apparently by a special mechanism (6-8). Reasons for the tight control of the frequency of replication initiation include constraints imposed by a multireplicon chromosomal organization, by complex mitotic mechanisms, and by the necessity to maintain constant gene dosages for at least some ofthe genes. Molecular mechanisms for suppression of extra rounds of DNA replication are unknown and may include specific chromatin structures at the replication origins or termini and possibly replication-dependent chemical modifications of DNA segments involved in replication initiation.The important point, however, is that the probability of an extra ("illegitimate") round of DNA replication (called below "replicon misfiring") within any given chromosomal domain, although generally very low, is not zero, as indicated by the following lines of evidence.