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Genetic progression and the waiting time to cancer
Niko Beerenwinkel, Tibor Antal, David Dingli, Arne Traulsen, Kenneth W. Kinzler, Victor E. Velculescu, Bert Vogelstein, Martin A. Nowak
TL;DR
Cancer progression can follow many mutational pathways, yet the timing of tumor emergence is heterogeneous. The paper models colorectal cancer evolution with a Wright-Fisher process and finds that waiting time is strongly shaped by mutation-conferred fitness, decreasing roughly as 1/s.
Problem
Cancer can follow many mutational pathways, and researchers seek to explain heterogeneity in the time required for tumor development.
Method
The paper models somatic colorectal-cancer evolution with a Wright-Fisher process, extensive simulations, and analytical approximations of expected cancer waiting time.
Results
Average tumor waiting time is strongly affected by mutation-conferred fitness and decreases roughly as 1/s.
Takeaways & Limitations
The model offers a basic understanding of how selection shapes the waiting time for tumor appearance.
Takeaways & Limitations
The study motivating the model examined a limited number of mutations, and gene interactions may depend on genetic background.
Abstract
from arXiv · showhide
Cancer results from genetic alterations that disturb the normal cooperative behavior of cells. Recent high-throughput genomic studies of cancer cells have shown that the mutational landscape of cancer is complex and that individual cancers may evolve through mutations in as many as 20 different cancer-associated genes. We use data published by Sjoblom et al. (2006) to develop a new mathematical model for the somatic evolution of colorectal cancers. We employ the Wright-Fisher process for exploring the basic parameters of this evolutionary process and derive an analytical approximation for the expected waiting time to the cancer phenotype. Our results highlight the relative importance of selection over both the size of the cell population at risk and the mutation rate. The model predicts that the observed genetic diversity of cancer genomes can arise under a normal mutation rate if the average selective advantage per mutation is on the order of 1%. Increased mutation rates due to genetic instability would allow even smaller selective advantages during tumorigenesis. The complexity of cancer progression thus can be understood as the result of multiple sequential mutations, each of which has a relatively small but positive effect on net cell growth.
Introduction 1
Colorectal cancers can contain many mutations and follow diverse mutational pathways, motivating a Wright-Fisher model of cancer progression. The model examines waiting time as a function of population size, mutation rate, and selective advantage, using simulations and analytical approximations.
- Genomic complexity: Individual colorectal tumors contained an average of 62 nonsynonymous mutations.Sjöblom et al. sequenced 13,000 genes in colorectal cancers.
- Genomic complexity: About 100 nonsynonymous mutations and up to 20 potentially causal mutated genes may occur in individual colorectal cancers.These estimates were extrapolated to the entire genome.
- Model motivation: Diverse mutational patterns among patients suggest that many mutational pathways can lead to the same cancer phenotype.The model therefore assumes 100 potential driver genes and considers acquisition of mutations in up to 20 genes.
- Model framework: The Wright-Fisher process models somatic cancer evolution in a colonic adenoma.The analysis assumes one cell turnover per day and varies population size N, per-gene mutation rate u, and average selective advantage s per mutation.
- Study contribution: The study combines extensive simulations with analytical approximations to estimate expected waiting time and assess how evolutionary forces contribute to cancer progression.The stated evolutionary parameters are population size, mutation rate, and selective advantage.
Methods 19
The study combines colorectal cancer mutation data with Wright-Fisher simulations and an analytical approximation to model stochastic accumulation of driver mutations. The framework represents genotypes by driver-gene mutation classes and estimates waiting times to the first k-fold mutant.
- Data: ~13,000 genes were sequenced from cancers of 11 patients with advanced colorectal cancers, and mutant genes were analyzed in an additional 24 patients.MMR-deficient tumors were excluded because mismatch repair deficiency increases mutation rates by orders of magnitude.
- Data: Mutations were found in 519 genes, including 105 genes mutated in at least two independent tumors.These data informed the cancer progression model and candidate driver analysis.
- Statistical analysis: The analysis calculated all 3003 pair-wise partial correlations among 78 candidate driver genes using a shrinkage estimation method.Shrinkage was used because the number of observed tumors was much smaller than the number of genes.
- Wright-Fisher process: Because the target is the first k-fold mutant with k = 20, simulations track the k + 1 mutant error classes rather than all 2^100 possible mutants.Each additional mutation receives selective advantage s, and back mutation is ignored.
- Analytical approximation: The analytical approximation decouples mutation and selection, models mutant classes as a traveling Gaussian wave, and estimates waiting time to the first k-fold mutant.The replicator equation underestimates waiting time because high-fitness higher-order mutants are generated instantaneously; the stochastic model accounts for their delayed expansion.
Results 5
Colorectal tumors show highly diverse mutational patterns, with a few frequently mutated genes and many genes altered only in small tumor subsets. Wright-Fisher simulations and an analytical approximation show that positive selection strongly shapes the waiting time to cancer, while population size and mutation rate provide important tradeoffs.
- Mutational diversity: Tumors harbor 1 to 20 mutated genes, with a mean of 6.5, and 66/78 = 85% of candidate genes mutated in at most 3 tumors.APC, p53, and K-ras were mutated in 24, 17, and 16 tumors, respectively.
- Mutational diversity: Most genes involved in tumor progression are mutated in only small subsets of tumors, without a clear overall pattern.A very small number of genes are frequent across tumors, whereas many others contribute to progression sporadically.
- Waiting time to cancer: The expected time to cancer increases as cell population size, selective advantage, or mutation rate decreases.Progression is therefore slow when only a small subset of actively replicating stem cells is at risk, whereas genetic instability accelerates it.
- Waiting time to cancer: 5 to 15 years is sufficient for cancer development in adenomas of size 107 to 109 cells with mutation rate 10−7 per gene per cell division and 1% selective advantage per mutation.With mutation rate 10−5 per gene per cell division, populations of 105 to 107 cells and 0.1% selective advantage can reach the required mutations in the same interval.
- Analytical approximation: The analytical approximation is linear in k, closely matches Wright-Fisher simulations when s > 0, and explicitly highlights selective advantage as the dominant evolutionary force.The approximation also gives an explicit tradeoff among the evolutionary forces affecting tumorigenesis.
Discussion 9
The model indicates that selection has a stronger effect on cancer waiting time than mutation rate or population size, while stochastic, sequential mutations generate tumor heterogeneity. Its simplifying assumptions make predicted waiting times lower bounds and may not capture epistasis or mutation-order constraints.
- Model limitations: The model assumes equal incremental fitness effects and that any 20 driver-gene mutations produce cancer, making its waiting times lower bounds.In reality, epistasis can be positive or negative, constrain mutation order, and make specific combinations necessary for the cancer phenotype.
- Lesion growth and mutation accumulation: Small initiating lesions cannot produce cancer through 20 driver mutations with only small fitness advantages; larger early advantages may allow growth before smaller advantages accumulate.Once the lesion reaches an intermediate size, mutations with small fitness advantages can eventually convert it into cancer.
- Waiting time and evolutionary forces: Waiting time decreases roughly as 1/s with mutation-conferred fitness s, whereas mutation rate and population size affect it only logarithmically.The model therefore assigns selection a stronger influence on waiting time than mutation rate or population size.
- Tumor heterogeneity: Because mutations occur stochastically and each cancer has a unique complement of mutations, tumor heterogeneity can determine invasion, metastasis, and therapy resistance.The model treats biological heterogeneity as a direct consequence of tumorigenesis.