Modes of Selection in Tumors as Reflected by Two Mathematical Models and Site Frequency Spectra

Abstract

The tug-of-war model was developed in a series of papers of McFarland and co-authors to account for existence of mutually counteracting rare advantageous driver mutations and more frequent slightly deleterious passenger mutations in cancer. In its original version, it was a state-dependent branching process. Because of its formulation, the tug-of-war model is of importance for tackling the problem as to whether evolution of cancerous tumors is “Darwinian” or “non-Darwinian.” We define two Time-Continuous Markov Chain versions of the model, including identical mutation processes but adopting different drift and selection components. In Model A, drift and selection process preserves expected fitness whereas in Model B it leads to non-decreasing expected fitness. We investigate these properties using mathematical analysis and extensive simulations, which detect the effect of the so-called drift barrier in Model B but not in Model A. These effects are reflected in different structure of clone genealogies in the two models. Our work is related to the past theoretical work in the field of evolutionary genetics, concerning the interplay among mutation, drift and selection, in absence of recombination (asexual reproduction), where epistasis plays a major role. Finally, we use the statistics of mutation frequencies known as the Site Frequency Spectra (SFS), to compare the variant frequencies in DNA of sequenced HER2+ breast cancers, to those based on Model A and B simulations. The tumor-based SFS are better reproduced by Model A, pointing out a possible selection pattern of HER2+ tumor evolution. To put our models in context, we carried out an exploratory study of how publicly accessible data from breast, prostate, skin and ovarian cancers fit a range of models found in the literature.

Comparison of Tug-of-War Models Assuming Moran versus Branching Process Population Dynamics

“…While in Moran A fitness stays constant (as expected), in the case of Moran B, clones with higher fitness are favored, even for the same initial conditions and in absence of new mutations (see section 3.1.1 in [16]). This behavior results from the difference between Moran A and Moran B in the expected change in population fitness after a death-replacement event.…”

“…As noted in [19], the most important difference is that of the expected value of fitness increment in the population at the moment of death - replacement in Model A versus Model B. The fitness increment is equal to the difference f j − f i , where f i , f j are fitnesses of the dead cell and of the new cell, in the absence of mutations.…”

“…In essence, S ( x ) is an idealized version of the empirical variant allele frequency (VAF) graph. In addition, it is convenient for reasons explained in [19] (Section 2.4) to define the cumulative tail of the SFS S ( x ) The theory that allows computing the expectations of SFS in populations with a given growth law under the Infinite Site Model (ISM) of mutation, was developed concurrently by many researchers, with one of the seminal papers published in 1998 by Griffiths and Tavaré [15]. The Griffiths-Tavaré expressions are accurate but quite complicated.…”

“…The original tug-of-war model [23] assumed that the cell death rate increases with the number of cells in the population increasing, which creates a mechanism for limiting the eventual tumor size. In other papers [19, 20], a Moran model was used for the population process, which provides a strict bound on the number of proliferating cells (see relevant discussion in [19]). Another assumption of [23] was that driver mutations become instantly fixed in the population, which may be acceptable under very strong selection (for mathematical details, see Bobrowski et al [2]), but in general it is not satisfied.…”

“…In other papers [16,17], a Moran model was used for the population process, which provides a strict bound on the number of proliferating cells (see relevant discussion in [16]). Another assumption of [18] was that driver mutations become instantly fixed in the population, which may be acceptable under very strong selection (for mathematical details, see Bobrowski et al [2]), but in general it is not satisfied.…”