Population dynamics under gradual vs punctuated mutagenesis.

(A) Evidence for punctuated APOBEC mutagenesis (Petljak et al. 2019) from successive cell culture expansions, seeded with single progenitors from the preceding expansion. Sequencing of the expanded populations revealed large fluctuations in APOBEC-associated mutagenic signatures SBS2 and SBS13. (B) Evidence for punctuated copy-number evolution (Gao et al. 2017, Minussi et al. 2021). Patterns in branch lengths of reconstructed phylogenetic trees reveal fluctuations in the rates of copy number alterations. Dynamics prior to most recent common ancestor (MRCA) are unidentifiable. (C) Gradual vs. punctuated evolution. In gradual evolution novelty emerges and spreads at a constant rate over time. In punctuated evolution novelty emerges and spreads during distinct burst phases. (D) Fitness schematic of two 1-step adaptations which each independently confer a fitness advantage. (E) Fitness schematic for a 2-step adaptation, in which carrying one mutation confers a fitness disadvantage but carrying two mutations confers a fitness advantage. (F,G) Schematic of possible evolutionary dynamics for two 1-step adaptations (F) and of modes of valley-crossing with or without prior fixation of the first mutant (G). Sequential fixation occurs at low mutation rates where emerging mutant lineages are likely to have fixated or gone extinct before the next mutation occurs. At higher mutation rates, the second mutation can occur in a multi-clonal population. (H) Sketch of failing two-step adaptation under a uniform (time-invariant) mutation rate. (I) Sketch of a successful two-step adaptation via stochastic tunneling under a punctuated mutation process with distinct clusters of high mutation rates

Simulation results: valley-crossing under uniform vs. temporally clustered mutation rates.

(A) Schematic of a Wright Fisher Process. (B)-(C) Fitness landscapes used for the simulations in Panel (D) and (E) respectively. Mutations move a cell from left to right through the landscape. Each 2-step adaptation corresponds to a fitness increase by a factor of 1.5. Having an odd number of mutations comes at a multiplicative fitness disadvantage of 0.5 in (B) and an advantage of 1.01 in panel (D). (D)-(E) Simulation results for a Wright-Fisher process with 50 cells. The mutation rate trajectories in each panel are chosen such that the total expected number of mutations under the uniform trajectory is identical to that under the temporally clustered trajectory.

Exploration and exploitation with temporally clustered mutation rates.

(A) Cells move through two-dimensional fitness landscapes. These fitness landscapes are randomly redrawn every 50714 divisions. (B) Mutation rate trajectories are parameterized with a mean mutation rate μ, and a clustering parameter k. (C) Average fitness in simulations of a population of 20 cells in a Wright Fisher Process. Simulations were run for 108 division events. (D)-(F) Average fitness trajectories for representative snippets of the simulations in (C).

Proxies for valley crossing and temporal clustering in simulations and in TCGA data.

(A) Sketch of the simulation analysis workflow. (B) Distributions of the fraction of mutations acquired during mutation bursts and the TSG deactivation score in simulations. (C) Joint distribution of the quantities in (B) indicates strong correlation. (D) Schematic of analysis workflow for TCGA data (methods) and results for the four cancer types with highest APOBEC signature contribution. (E) Probabilities that single base substitutions (SBSs) in samples of the cancer types in (C) were caused by APOBEC associated signatures SBS2 and SBS13. On the left, these probabilities are averaged over all SBSs. On the right, only those SBSs are considered which are classified as “nonsense” or “missense”, and which appear in a TSG with at least 2 such deactivating SBSs. The average probabilities between both groups are significantly different (t-test, p<0.001). (F) Average probabilities analogous to those in panel (D) were constructed for each individual sample and then averaged across samples. Samples without deactivated TSGs were excluded. The average probabilities between groups are significantly different for the four cancer types (*, ** and *** respectively indicate that the p-value in a t-test lies below 0.05, 0.01 and 0.001).

Stochastic tunneling vs clonal interference under temporally clustered mutation rates:

(A) Schematic of the simulation approach. (B) Sketch of the mutation process. (C) Simulation results: measuring fixations of higher-fitness mutations per generation as a function of the clustering parameter and of the fitness landscape. In the 1-step adaptation setting fitness is defined as 1.5#mutations. Fitness in the 2-step adaptation setting is defined as in Fig 2B. Results are averaged over simulation runs with 107 generations. (D) Sketch of fitness landscape for simulations shown in (E). Cells start with an unmutated genome of 200 loci. Mutations on each locus have independent multiplicative effects on cell fitness. In the first 100 loci a single mutation confers a multiplicative fitness change of 1.5. In the remaining 100, 2 mutations are required to reach this multiplicative fitness change of 1.5, with the first mutation conferring a multiplicative fitness change of 0.5. (E) Simulation results for adaptation with genome sketched in (D). Results are averaged over 100 simulation runs per value of k. Shaded regions indicate 10th and 90th percentiles. Smaller plots (left) are zoomed in versions of the first 1200 generations of the larger plots (right), with identical color-coding.

Simulation results: valley-crossing under uniform vs. temporally clustered mutation rates in branching process.

(A) Schematic of fitness landscape as in Fig 2. (B) Schematic of Branching Process model. (C) Simulation results for the Branching Process model. The mutation rate trajectories in (C) are chosen such that the total expected number of mutations under the uniform trajectory is identical to that under the temporally clustered trajectory.

Effect of temporal clustering on stochastic tunneling rates as a function of the mutation rate.

(A) Sketch of the mutation process with clustering parameter k. (B) Sketch of the fitness landscape as in Fig 2. (C) Description of the simulations. (D)-(E) Simulation results of a Wright-Fisher process with 100 cells over 106 generations with a fixed average mutation rate of10−5per cell per generation and varying temporal clustering k. The fitness is as in (B), with subsequent peaks differing in fitness by a factor of 1.5. However, on the y-axis in (F) and (G) we vary the fitness of the valleys relative to the preceding peak from 0.5 (as in (B)) to 1.5. Panel (D) shows the number of valleys crossed per generation (color bar in log10 values), where contrary to the previous panels it is no longer true that every valley crossing leads to a fixation – the next mutant might emerge before that. Panel (E) the share of these valley crossings that occur without fixation of the first mutant.(F) Simulation results of the process described in (A)-(C). Each simulation ran for105/(kμ) generations. The y-axis measures fixations of adaptive mutations per generation (see Fig 5D). (G) Fixations per generation relative to the uniform mutation process (k=1) for k=5 and k=10.

TSG deactivation scores in TCGA.

Distribution and correlation of APOBEC signature contribution and TSG deactivation scores as in Fig 4, sorted by cancer type.

Synergistic mutation scores in TCGA.

Distribution and correlation of APOBEC signature contribution and synergistic mutation scores, sorted by cancer type.

Temporal clustering and adaptation in fitness landscapes with varying ruggedness.

(A) Sketch of the setup. Like in Fig 5, we assume that cells have two types of loci on which one-step or two-step adaptations are possible respectively. Fitness effects per locus are as in Fig 5. Here, we run simulations in which we vary the relative shares of the two types of loci. (B) Simulation results over time, averaged for 100 simulation replicates per row and clustering parameter k. Colors indicate the fitness ratio between simulations with k=5 and k=1 on a log10 scale. (C)-(D) show the corresponding absolute numbers of loci with adaptative mutations (i.e. loci in the black and blue states in panel (A)).