Fig 1.
Diagram of the generalized birth-and-death model of cancer evolution.
At each generation, each cell (X) can either die (red) or divide (blue) and produce two daughter cells (Y1 and Y2) with probabilities that depend on the death (D) and birth (B) rates of the system. For each daughter cell, the probability that it acquires +0, +1, +2, …, +k mutations (pk) is given by a Poisson distribution with mean μ(Td+Tp). The new mutations (kmut) are catalogued as drivers with probability or passengers with probability
. Driver mutations are randomly assigned to a cancer gene, that may or may not be involved in the epistasis network. The definitions of the parameters and the expressions of the birth and death rates are provided in the main text and Table 1.
Table 1.
Default parameter values used in the simulations.
Fig 2.
Synergistic epistasis can induce evolutionary rescue of receding tumors.
(A) Representative trajectory of a precancerous lesion subject to synergistic epistasis among drivers (green), superimposed to a representative trajectory without epistasis (brown). Parameter values: μ = 10−8, sd = 0.07, ϵ = 2, fe = 0.5, G = 70, N0 = 103. (B) Schematic of the dynamics of an initial receding tumor that is rescued by a trigger driver that appears after approximately 6000 cell divisions. Nc and represent the critical population sizes before and after the trigger event, respectively. For the rescue to occur, the trigger driver must appear before the population size drops below
.
Fig 3.
Tumor rescue in real-world epistasis networks.
(A) Simplified representation of regular (left) and star-like (right) networks. The actual star and clique networks consist of up to 70 genes. (B, from left to right) Structure of positive epistasis networks for several representative cancer subtypes; their degree distributions; comparison of the probability of tumor progression among regular (C), representative cancer subtype (R), and star-like (S) networks; and median triggering time in regular (green), representative cancer subtype (black), and star-like (brown) networks. For each cancer subtype, the rescue probability and trigger time of the empirical epistasis network are compared with regular and star-like networks with the same hub connectivity (see Methods). The gray bars in the degree distribution correspond to the fraction of nodes that are not affected by epistasis. Networks for representative cancer subtypes are based on the modular structure of empirical epistasis networks (the identity of the most connected cancer gene is indicated in each case). The triggering time is determined by the occurrence of the first driver that makes . Parameter values: N0 = 103, sp = 10−3, Tp = 5×106, ϵ = 6, μ = 5×10−9, variable Td according to the number of cancer genes associated with each cancer type.
Fig 4.
Tumor rescue is facilitated by high mutation rates and strong epistasis.
Probability of tumor progression as a function of network structure, mutation rate (μ), driver fitness effect (sd), fraction of genes subject to epistasis (fe), and strength of epistasis (ϵ). The probabilities were calculated for 1000 independent trajectories and each trajectory was classified as “tumor progression” if the population doubled its size before 30,000 cell divisions. The solid and dashed lines correspond to the analytical curves for a probability of 0.5 when taking and not taking into account the upper time limit of 30,000 cell divisions, respectively. In the region between the two curves, tumor progression is technically possible, but too slow to become manifest during human lifespan. Parameter values: N0 = 103, sp = 10−3, Tp = 5×106, Td = 700, G = 70.
Fig 5.
Timing of tumor rescue is critically affected by the structure of the epistasis network.
(A) Distribution of tumor rescue times (understood as the time of appearance of a trigger driver in the most connected gene) in star- (brown) and clique-like (green) networks, and their variation with the mutation rate (μ), driver fitness effect (sd), fraction of genes subject to epistasis (fe), and strength of epistasis (ϵ). Histograms were obtained from 1000 independent trajectories; solid lines correspond to analytical expressions. To facilitate comparisons, all the distributions are normalized to the same total area. (B) Dependence of the median rescue times (solid lines) and their 25–75 percentiles (shaded areas) on the parameters of the model, based on analytical expressions. Parameter values: μ = 5×10−9, sd = 0.05, fe = 0.5, ϵ = 2, G = 70, rest of parameters as in Fig 2B.
Fig 6.
Predictability of tumor progression given the mutational profile.
The number of drivers and passengers was collected at every time step for 1000 independent trajectories. For each combination with >10 observations, the figure indicates the fraction of trajectories that ended up doubling the initial size after 30,000 cell divisions. The dashed and solid lines correspond to the critical driver-to-passenger ratios (Rc and ) that determine the fate of the cell population before and after the trigger event. In the absence of epistasis
(bottom plot). Parameter values: μ = 10−8, sd = 0.07, fe = 0.5, G = 70, rest of parameters as in Fig 2B.
Fig 7.
Number of drivers acquired by clones that do not progress to cancer.
The distributions correspond to the maximum number of drivers observed along 1000 trajectories that did not double their initial size in 30,000 cell divisions, considering a clique-like epistasis network (see S2 Fig for a star-like network). Parameter values: μ = 5×10−9, sd = 0.05, fe = 0.5, G = 70, ϵ = 2, rest of parameters as in Fig 2B.