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To the Editor — Williams et al.1 analyzed next-generation sequencing data from bulk tumor samples, supporting the hypothesis that selection is limited to times before malignant transformation while tumor cell populations afterward evolve exclusively by neutral evolution. The authors arrived at their conclusions by showing that the expected number of mutations (M) in an exponentially increasing population undergoing only neutral evolution grows linearly with the inverse allele frequency (1/f) of mutant alleles. The allele frequency of a variant is defined as the proportion of cells containing that particular variant. Their reasoning was based on the fact that, in a neutrally evolving cell population, all subclones grow at the same rate, so the allele frequency is fixed as the inverse of the number of cells present at the time of appearance of a new variant. The authors then concluded that a tumor displaying a linear relationship between the number of mutations and the inverse allele frequency should imply that the population evolves without selection. They also performed simulations of a branching-process model with selection to show that selection cannot explain a linear relationship between M and 1/f. Furthermore, they observed a high correlation between M and 1/f in about one-third of the patient samples investigated as further indication of neutral tumor evolution. Their findings corroborate previous results2 based on the ‘Big Bang’ model that suggested a similar conclusion in colorectal cancer using single-time-point bulk sequencing data. We believe that the authors arrived at an erroneous conclusion, which also stands in contrast to other recent findings in this field3–5, based on flawed logic known as the ‘fallacy of the converse’. The fact that a model of neutral evolution leads to a linear relationship between M and 1/f does not imply that a linear relationship proves the presence of neutral evolution. In more abstract terms, A implying B does not necessarily mean that B implies A. Here we demonstrate that models with selection can also lead to a linear relationship between M and 1/f and that, therefore, linearity is a test statistic that cannot be used to distinguish between populations evolving with and without selection. Our results indicate that the claims made by Williams et al.1 have little merit. We designed and analyzed two alternative stochastic evolutionary models that both return a linear relationship and show similarly high R2 values for neutral as well as selection scenarios. To be consistent with the assumptions of the model and results used by Williams et al.1, both models are based on exponentially growing, noncompeting cellular populations with no spatial or microenvironmental effects and thus represent, by design, simplified versions of the tumorigenic process. The first model is a simple birth–death process of mutation accumulation (Fig. 1). In this model, each new mutation event gives rise to a single variant allele. This approach allows derivation of exact expressions for R2 = 0.986 R2 = 0.987 R2 of linear gression

Currently available bulk sequencing data do not necessarily support a model of neutral tumor evolution