Fig 1.
Evolutionary dynamics of passenger mutations during clonal expansion.
(A) New passenger mutations can be lost due to stochastic drift (diamonds). Successful mutations form surviving lineages. We order successful mutations by their time of appearance. Individual cells can harbor many passenger mutations and various different phylogenies can arise (B). In the example shown, mutation 2 appears in a cell that already harbors mutation 1. Thus all cells that have mutation 2 also have mutation 1. Similarly, all cells cells that have mutations 4 or 5 also harbor mutations 1 and 2. Mutation 3 forms an independent clone. We calculate the likelihood of different phylogenies and the expected number of subclonal mutations of any frequency.
Fig 2.
Frequency of passenger mutations.
(A-B) Cumulative distribution function for the first three successful mutations. The y-axis shows the probability that the mutation has a frequency of less than α. Comparison between formula (2) and exact computer simulations of the stochastic process with death-birth ratios δ = 0.72 (A) and δ = 0.99 (B). For δ = 0.72, the median frequencies of the first three successful mutations are below 5%. For δ = 0.99, they are all above 40%. (C-D) Mutation frequency versus time of appearance. (C) Mean frequency attained by a mutation which arose when there were z other cells in the population, for different values of the death-birth ratio, δ. (D) Maximum likelihood and maximum a posteriori estimate (which are approximately equal) for the number of cells in the population when the mutation with frequency α arose. Passenger mutation rate u = 0.015 (product of the number of basepairs in the exome, L ∼ 3 ⋅ 107, and the normal point mutation rate during cell division, μ ∼ 5 ⋅ 10−10).
Fig 3.
Likelihood of phylogenetic trees.
(A) All six phylogenetic trees containing the first three surviving passenger mutations are shown. (B) Probabilities of each tree for different values of the death-birth ratio, δ (formulas shown in Methods). For δ = 0.72, the first tree is the most likely. For δ = 0.99, the sixth tree is the most likely. For intermediate δ = 0.97, the most likely tree shape is that of trees 2-4. Passenger mutation rate u = 0.015.
Table 1.
Expected number of subclonal and clonal mutations for different values of δ = d/b.
Fig 4.
Predicted and observed numbers of subclonal mutations in colorectal cancer.
Exome sequencing data for two colorectal cancers from the TCGA dataset, (A) microsatellite stable (MSS) and (B) microsatellite instable (MSI), show the corrected allele fraction of each detected mutation (observed allele fraction divided by purity). Mutations with allele frequency of 25% or more may be clonal [34] and mutations with corrected allele frequency below 12% can be difficult to detect reliably. Thus we focus on mutations with fractions between 0.12 and 0.25, and plot the number of mutations with fraction between α and 0.25 as a function of α. The data are fit to the formula for the number of mutations with the corresponding allele frequency Eq (7). The best fit for and its corresponding 95% confidence interval is shown for each sample.
Table 2.
Likelihood of phylogenetic trees.