SUMMARYCancers arise through a process of somatic evolution that can result in substantial sub-clonal heterogeneity within tumors. The mechanisms responsible for the coexistence of distinct sub-clones and the biological consequences of this coexistence remain poorly understood. Here we used a mouse xenograft model to investigate the impact of sub-clonal heterogeneity on tumor phenotypes and the competitive expansion of individual clones. We found that tumor growth can be driven by a minor cell subpopulation, which enhances the proliferation of all cells within a tumor by overcoming environmental constraints and yet can be outcompeted by faster proliferating competitors, resulting in tumor collapse. We then developed a mathematical modeling framework to identify the rules underlying the generation of intratumor clonal heterogeneity. We found that non-cell autonomous driving, together with clonal interference, stabilizes sub-clonal heterogeneity, thereby enabling inter-clonal interactions that can lead to new phenotypic traits.

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

Weak selection, which means a phenotype is slightly advantageous over another, is an important limiting case in evolutionary biology. Recently, it has been introduced into evolutionary game theory. In evolutionary game dynamics, the probability to be imitated or to reproduce depends on the performance in a game. The influence of the game on the stochastic dynamics in finite populations is governed by the intensity of selection. In many models of both unstructured and structured populations, a key assumption allowing analytical calculations is weak selection, which means that all individuals perform approximately equally well. In the weak selection limit many different microscopic evolutionary models have the same or similar properties. How universal is weak selection for those microscopic evolutionary processes? We answer this question by investigating the fixation probability and the average fixation time not only up to linear but also up to higher orders in selection intensity. We find universal higher order expansions, which allow a rescaling of the selection intensity. With this, we can identify specific models which violate ͑linear͒ weak selection results, such as the one-third rule of coordination games in finite but large populations.

Abstract. In evolutionary game dynamics, reproductive success increases with the performance in an evolutionary game. If strategy A performs better than strategy B, strategy A will spread in the population. Under stochastic dynamics, a single mutant will sooner or later take over the entire population or go extinct. We analyze the mean exit times (or average fixation times) associated with this process. We show analytically that these times depend on the payoff matrix of the game in an amazingly simple way under weak selection, i.e. strong stochasticity: the payoff difference π is a linear function of the number of A individuals i, π = u i + v. The unconditional mean exit time depends only on the constant term v. Given that a single A mutant takes over the population, the corresponding conditional mean exit time depends only on the density dependent term u. We demonstrate this finding for two commonly applied microscopic evolutionary processes.

Underdominance refers to natural selection against individuals with a heterozygous genotype. Here, we analyse a single-locus underdominant system of two large local populations that exchange individuals at a certain migration rate. The system can be characterized by fixed points in the joint allele frequency space. We address the conditions under which underdominance can be applied to transform a local population that is receiving wildtype immigrants from another population. In a single population, underdominance has the benefit of complete removal of genetically modified alleles (reversibility) and coexistence is not stable. The two population system that exchanges migrants can result in internal stable states, where coexistence is maintained, but with additional release of wildtype individuals the system can be reversed to a fully wildtype state. This property is critically controlled by the migration rate. We approximate the critical minimum frequency required to result in a stable population transformation. We also concentrate on the destabilizing effects of fitness and migration rate asymmetry. Practical implications of our results are discussed in the context of utilizing underdominance to genetically modify wild populations. This is of importance especially for genetic pest management strategies, where locally stable and potentially reversible transformations of populations of disease vector species are of interest.

The idea of introducing genetic modifications into wild populations of insects to stop them from spreading diseases is more than 40 years old. Synthetic disease refractory genes have been successfully generated for mosquito vectors of dengue fever and human malaria. Equally important is the development of population transformation systems to drive and maintain disease refractory genes at high frequency in populations. We demonstrate an underdominant population transformation system in Drosophila melanogaster that has the property of being both spatially self-limiting and reversible to the original genetic state. Both population transformation and its reversal can be largely achieved within as few as 5 generations. The described genetic construct {Ud} is composed of two genes; (1) a UAS-RpL14.dsRNA targeting RNAi to a haploinsufficient gene RpL14 and (2) an RNAi insensitive RpL14 rescue. In this proof-of-principle system the UAS-RpL14.dsRNA knock-down gene is placed under the control of an Actin5c-GAL4 driver located on a different chromosome to the {Ud} insert. This configuration would not be effective in wild populations without incorporating the Actin5c-GAL4 driver as part of the {Ud} construct (or replacing the UAS promoter with an appropriate direct promoter). It is however anticipated that the approach that underlies this underdominant system could potentially be applied to a number of species.

The environment in which a population evolves can have a crucial impact on selection. We study evolutionary dynamics in finite populations of fixed size in a changing environment. The population dynamics are driven by birth and death events. The rates of these events may vary in time depending on the state of the environment, which follows an independent Markov process. We develop a general theory for the fixation probability of a mutant in a population of wild-types, and for mean unconditional and conditional fixation times. We apply our theory to evolutionary games for which the payoff structure varies in time. The mutant can exploit the environmental noise; a dynamic environment that switches between two states can lead to a probability of fixation that is higher than in any of the individual environmental states. We provide an intuitive interpretation of this surprising effect. We also investigate stationary distributions when mutations are present in the dynamics. In this regime, we find two approximations of the stationary measure. One works well for rapid switching, the other for slowly fluctuating environments.

On studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspirationdriven dynamics. In the former case, individuals imitate the strategy of a more successful peer. In the latter case, individuals adjust their strategies based on a comparison of their pay-offs from the evolutionary game to a value they aspire, called the level of aspiration. Unlike imitation processes of pairwise comparison, aspiration-driven updates do not require additional information about the strategic environment and can thus be interpreted as being more spontaneous. Recent work has mainly focused on understanding how aspiration dynamics alter the evolutionary outcome in structured populations. However, the baseline case for understanding strategy selection is the well-mixed population case, which is still lacking sufficient understanding. We explore how aspiration-driven strategy-update dynamics under imperfect rationality influence the average abundance of a strategy in multi-player evolutionary games with two strategies. We analytically derive a condition under which a strategy is more abundant than the other in the weak selection limiting case. This approach has a long-standing history in evolutionary games and is mostly applied for its mathematical approachability. Hence, we also explore strong selection numerically, which shows that our weak selection condition is a robust predictor of the average abundance of a strategy. The condition turns out to differ from that of a wide class of imitation dynamics, as long as the game is not dyadic. Therefore, a strategy favoured under imitation dynamics can be disfavoured under aspiration dynamics. This does not require any population structure, and thus highlights the intrinsic difference between imitation and aspiration dynamics.