The mechanics of stochastic slowdown in evolutionary games

We study the stochastic dynamics of evolutionary games, and focus on the so-called `stochastic slowdown' effect, previously observed in (Altrock et. al, 2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the prisoner's dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the microscopic origins of these phenomena, we calculate the average sojourn times. This allows us to identify the transient states which contribute most to the slowdown effect, and enables us to provide an understanding of slowdown in the takeover of a small group of cooperators by defectors: Defection spreads quickly initially, but the final steps to takeover can be delayed substantially. The analysis of coexistence games reveals even more intricate behavior. In small populations, the conditional average fixation time can show multiple extrema as a function of the selection strength, e.g., slowdown, speedup, and slowdown again. We classify two-player games with respect to the possibility to observe non-monotonic behavior of the conditional average fixation time as a function of selection strength.Comment: Accepted for publication in the Journal of Theoretical Biology. Includes changes after peer revie

Aspiration Dynamics of Multi-player Games in Finite Populations

Studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspiration-driven 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 payoffs 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 game 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 favored under imitation dynamics can be disfavored under aspiration dynamics. This does not require any population structure thus highlights the intrinsic difference between imitation and aspiration dynamics

Fixation in finite populations evolving in fluctuating environments

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

Nonequilibrium phase transitions in finite arrays of globally coupled Stratonovich models: Strong coupling limit

A finite array of $N$ globally coupled Stratonovich models exhibits a continuous nonequilibrium phase transition. In the limit of strong coupling there is a clear separation of time scales of center of mass and relative coordinates. The latter relax very fast to zero and the array behaves as a single entity described by the center of mass coordinate. We compute analytically the stationary probability and the moments of the center of mass coordinate. The scaling behaviour of the moments near the critical value of the control parameter $a_c(N)$ is determined. We identify a crossover from linear to square root scaling with increasing distance from $a_c$. The crossover point approaches $a_c$ in the limit $N \to \infty$ which reproduces previous results for infinite arrays. The results are obtained in both the Fokker-Planck and the Langevin approach and are corroborated by numerical simulations. For a general class of models we show that the transition manifold in the parameter space depends on $N$ and is determined by the scaling behaviour near a fixed point of the stochastic flow

Development of a scoring function for comparing simulated and experimental tumor spheroids

Progress continues in the field of cancer biology, yet much remains to be unveiled regarding the mechanisms of cancer invasion. In particular, complex biophysical mechanisms enable a tumor to remodel the surrounding extracellular matrix (ECM), allowing cells to invade alone or collectively. Tumor spheroids cultured in collagen represent a simplified, reproducible 3D model system, which is sufficiently complex to recapitulate the evolving organization of cells and interaction with the ECM that occur during invasion. Recent experimental approaches enable high resolution imaging and quantification of the internal structure of invading tumor spheroids. Concurrently, computational modeling enables simulations of complex multicellular aggregates based on first principles. The comparison between real and simulated spheroids represents a way to fully exploit both data sources, but remains a challenge. We hypothesize that comparing any two spheroids requires first the extraction of basic features from the raw data, and second the definition of key metrics to match such features. Here, we present a novel method to compare spatial features of spheroids in 3D. To do so, we define and extract features from spheroid point cloud data, which we simulated using Cells in Silico (CiS), a high-performance framework for large-scale tissue modeling previously developed by us. We then define metrics to compare features between individual spheroids, and combine all metrics into an overall deviation score. Finally, we use our features to compare experimental data on invading spheroids in increasing collagen densities. We propose that our approach represents the basis for defining improved metrics to compare large 3D data sets. Moving forward, this approach will enable the detailed analysis of spheroids of any origin, one application of which is informing in silico spheroids based on their in vitro counterparts. This will enable both basic and applied researchers to close the loop between modeling and experiments in cancer research

Universality of weak selection

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 (linear) weak selection results, such as the one--third rule of coordination games in finite but large populations.Comment: 12 pages, 3 figures, accepted for publication in Physical Review

Fixation times in evolutionary games under weak selection

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, ie strong stochasticity: The payoff difference $\Delta \pi$ is a linear function of the number of $A$ individuals $i$, $\Delta \pi = u i + v$. 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.Comment: Forthcoming in New Journal of Physic

Stochastic slowdown in evolutionary processes

We examine birth--death processes with state dependent transition probabilities and at least one absorbing boundary. In evolution, this describes selection acting on two different types in a finite population where reproductive events occur successively. If the two types have equal fitness the system performs a random walk. If one type has a fitness advantage it is favored by selection, which introduces a bias (asymmetry) in the transition probabilities. How long does it take until advantageous mutants have invaded and taken over? Surprisingly, we find that the average time of such a process can increase, even if the mutant type always has a fitness advantage. We discuss this finding for the Moran process and develop a simplified model which allows a more intuitive understanding. We show that this effect can occur for weak but non--vanishing bias (selection) in the state dependent transition rates and infer the scaling with system size. We also address the Wright-Fisher model commonly used in population genetics, which shows that this stochastic slowdown is not restricted to birth-death processes.Comment: 8 pages, 3 figures, accepted for publicatio

First Steps towards Underdominant Genetic Transformation of Insect Populations

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. Figure

Deterministic evolutionary game dynamics in finite populations

Evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals. Typically, the microscopic definition of strategy spreading is stochastic, such that the dynamics becomes deterministic only in infinitely large populations. Here, we introduce a new microscopic birth--death process that has a fully deterministic strong selection limit in well--mixed populations of any size. Additionally, under weak selection, from this new process the frequency dependent Moran process is recovered. This makes it a natural extension of the usual evolutionary dynamics under weak selection. We find simple expressions for the fixation probabilities and average fixation times of the new process in evolutionary games with two players and two strategies. For cyclic games with two players and three strategies, we show that the resulting deterministic dynamics crucially depends on the initial condition in a non--trivial way.Comment: 11 pages, 7 figure