Stochastic Tunneling and Metastable States During the Somatic Evolution of Cancer

Abstract

Tumors initiate when a population of proliferating cells accumulates a certain number and type of genetic and/or epigenetic alterations. The population dynamics of such sequential acquisition of (epi)genetic alterations has been the topic of much investigation. The phenomenon of stochastic tunneling, where an intermediate mutant in a sequence does not reach fixation in a population before generating a double mutant, has been studied using a variety of computational and mathematical methods. However, the field still lacks a comprehensive analytical description since theoretical predictions of fixation times are available only for cases in which the second mutant is advantageous. Here, we study stochastic tunneling in a Moran model. Analyzing the deterministic dynamics of large populations we systematically identify the parameter regimes captured by existing approaches. Our analysis also reveals fitness landscapes and mutation rates for which finite populations are found in long-lived metastable states. These are landscapes in which the final mutant is not the most advantageous in the sequence, and resulting metastable states are a consequence of a mutationselection balance. The escape from these states is driven by intrinsic noise, and their location affects the probability of tunneling. Existing methods no longer apply. In these regimes it is the escape from the metastable states that is the key bottleneck; fixation is no longer limited by the emergence of a successful mutant lineage. We used the so-called Wentzel-Kramers-Brillouin method to compute fixation times in these parameter regimes, successfully validated by stochastic simulations. Our work fills a gap left by previous approaches and provides a more comprehensive description of the acquisition of multiple mutations in populations of somatic cells.KEYWORDS stochastic modeling; population genetics; cancer; Moran process; WKB method U NDERSTANDING the dynamics of an evolving population structure has long been the goal of population genetics. Several authors have constructed probabilistic models to study allele frequency distributions in populations subject to mutation, selection, and genetic drift (Fisher 1930;Wright 1931;Moran 1962). The mathematical analysis of these models leads to an improved understanding of the underlying system and has been crucial for the interpretation of the laws of evolution. This is most evident in the quantitative analysis of cancer, which has seen numerous studies throughout the 20th century that addressed the kinetics of cancer initiation and progression (Nordling 1953;Armitage and Doll 1954;Fisher 1958;Knudson 1971;Moolgavkar 1978). Due to these and other studies (see Weinberg 2013 for a review), we now know that human cancer initiates when cells within a proliferating tissue accumulate a certain number and type of genetic and/or epigenetic alterations. These alterations can be point mutations, amplification and deletion of genomic material, structural changes such as translocations, loss or gain of DNA methylation...

Effects of niche overlap on coexistence, fixation and invasion in a population of two interacting species

“…In the independent limit, a = 0, the invasion probability asymptotically approaches 1 for large K, reflecting the fact that the system is deterministically drawn towards the deterministic stable fixed point with equal numbers of both species. Interestingly, the invasion probability is a non-monotonic function of K and exhibits a minimum at an intermediate/low carrying capacity, which might be relevant for some biological systems, such as in early cancer development [13] or plasmid exchange in bacteria [17]. The explanation for this non-monotonicity is that increasing K has opposing effects: it makes the end goal of growing from one invader to half the population farther away, but it also increases the effective draw towards the deterministic fixed point.…”

“…Remarkable biodiversity exists in biomes such as the human microbiome [1-3], the ocean surface [4,5], soil [6], the immune system [7][8][9] and other ecosystems [10,11]. Accordingly, quantitative predictive understanding of the long-term population behaviour of complex populations is important for many human health and disease and industrial processes such as drug resistance in bacteria, cancer progression, evolutionary phylogeny inference algorithms and immune response [2,3,[12][13][14][15][16][17]. Nevertheless, the long-term dynamics, diversity and stability of communities of multiple interacting species are still incompletely understood.…”

The adaptive potential of non-heritable somatic mutations

“…Under these high germline mutation rates, populations increased in fitness by converging on the peak genotype via stochastic tunneling (Fig. 2C) [57]. When somatic mutations conferred a selective advantage to the organism, the mean fitness of the population increased with both the germline and somatic mutation rates, thus expanding the parameter space in which higher fitness evolved, relative to the control simulations (compare Figs.…”

Temporal order of mutations influences cancer initiation dynamics

“…Due to these theoretical studies and a large number of other important experimental studies, we currently know that the cancer starts when cells accumulate certain number and types of genetic alterations. Significant efforts have been made for modeling the dynamics of mutation acquisition and how it is governed by relevant genetic parameters such as the rate of mutations, the size of the population of cells and the rate of mutations proliferation [1,5,12,16,19,20]. Recently we developed a new theoretical framework to evaluate the cancer initiation dynamics [23].…”