As the rates of most cancers are proportional to the fourth to fifth power of age ("log-log" behavior), it is widely believed that 5 to 6 independent mutations are necessary for malignant transformation. Conversely, the peak incidences of most cancers are similar to stem-cell mutation rates at single loci, implying only one rate-limiting mutation. Here, flow cytometrically measured red blood cells mutated at a selectively neutral locus, glycophorin A, allow observation of individual stem-cell differentiation events in a loglog malignancy, polycythemia rubra vera. Contrary to predictions from multistep models, the clone is driven by infrequent (< annual) and rare (ϳ 18 per year) differentiation events. These parameters imply that malignant stem cells have a modest selective advantage. Correspondingly minor, typically less than 20%, increases in stochastic self-renewal ratios are modeled to show that single mutations can result in the observed fourth power relationship with age. The conundrum between log-log behavior and mutation rate is thereby reconcilable, with the age of onset arising not from the requirement for multiple, independent mutations but from infrequent, stochastic stem-cell division rates and single mutations causing initially minor effects, but initiating a clone whose expected number increases successively with age-an "exponential phe
IntroductionThe incidences of most human cancers increase as a fourth to fifth power of age. 1 Plotting logarithmically transformed data approximates to straight lines, giving rise to the term log-log cancers. As the probability of independent stochastic events occurring together in the same cell is the product of the probabilities of each event, and the corresponding rate is the first derivative with respect to time, 2,3 it is widely believed there are 5 to 6 rate-limiting mutations necessary to cause cancer. 4,5 However, mutation rates are many orders of magnitude too low to accommodate so many independent mutations, prompting 2-or 3-step theories. 6,7 Such models postulate that the first mutation alters phenotype, either proliferative 8 or mutator, 9,10 to make subsequent rate-limiting mutations more likely. However, none of these models has gained general acceptance, partly because measurements on malignant stem cells are lacking. 11,12 Polycythemia rubra vera (PRV) is a myeloproliferative, stem cell, clonal disorder, whose incidence increases as a fourth power of age with a peak incidence of 10.5 ϫ 10 5 /year at ages 75 to 80 years. 13 The disease is closely associated with a point mutation, 617VϾF, in the JAK2 gene, 14-18 although it is currently uncertain whether the mutation is primary or secondary. 19,20 In healthy individuals, about 3 ϫ 10 6 hematopoietic stem cells cycle approximately annually, 21 and point mutation rates are about 1 ϫ 10 Ϫ10 per cell division. The expected annual rate of production of new JAK2 617VϾF mutations in stem cells is therefore 3 ϫ 10 6 ϫ 10 Ϫ10 ϭ 3 ϫ 10 Ϫ4 per individual per year or 30 per 10 5 /year, 21 similar to the peak di...