A computational model of MGUS progression to Multiple Myeloma identifies optimum screening strategies and their effects on mortality
Altrock, P.M., et al., 2018. A computational model of MGUS progression to Multiple Myeloma identifies optimum screening strategies and their effects on mortality. JCO Clinical Cancer Informatics , 2018 (2018(2) , pp. 1-12. Publisher's VersionAbstract
PURPOSE: Recent advances uncovered therapeutic interventions that might reduce the risk of progression of premalignant diagnoses, such as Monoclonal Gammopathy of Undetermined Significance (MGUS) to multiple myeloma (MM). It remains unclear how to best screen populations at risk and how to evaluate the ability of these interventions to reduce disease prevalence and mortality at the population level. To address these questions, we developed a computational modeling framework. METHODS: We used individual-based computational modeling of MGUS incidence and progression across a population of diverse individuals, to determine best screening strategies in terms of screening start, intervals, and risk-group specificity. Inputs were life tables, MGUS incidence and baseline MM survival. We measured MM-specific mortality and MM prevalence following MGUS detection from simulations and mathematical precition modeling. RESULTS: We showed that our framework is applicable to a wide spectrum of screening and intervention scenarios, including variation of the baseline MGUS to MM progression rate and evolving MGUS, in which progression increases over time. Given the currently available progression risk-point estimate of 61% risk, starting screening at age 55 and follow-up screening every 6yrs reduced total MM prevalence by 19%. The same reduction could be achieved with starting age 65 and follow-up every 2yrs. A 40% progression risk reduction per MGUS patient per year would reduce MM-specific mortality by 40%. Generally, age of screening onset and frequency impact disease prevalence, progression risk reduction impacts both prevalence and disease-specific mortality, and screeenign would generally be favorable in high-risk individuals. CONCLUSION: Screening efforts should focus on specifically identified groups of high lifetime risk of MGUS, for which screening benefits can be significant. Screening low-risk MGUS individuals would require improved preventions.
Extinction rates in tumour public goods games
Gerlee, P. & Altrock, P.M., 2017. Extinction rates in tumour public goods games. Royal Society Interface , 14 (134). Publisher's VersionAbstract

Cancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumour cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the time scales, in particular, in coevolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell-type-specific rates have to be accounted for explicitly.

Evolutionary games on cycles with strong selection
Altrock, P.M., Traulsen, A. & Nowak, M.A., 2017. Evolutionary games on cycles with strong selection. Physical Review E , 95 , pp. 022407. Publisher's VersionAbstract


Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast in a dominance game, that fixation takes very long in a coexistence game, and that strong selection eliminates demographic noise. Here we show that these intuitions can be misleading in structured populations. We analyze mean fixation times on the cycle graph under strong frequency-dependent selection for two different microscopic evolutionary update rules (death-birth and birth-death). We establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size. Depending on the underlying game, we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring (death-birth update). In contrast, if selection for an offspring occurs before random removal (birth-death update), then strong selection can remove demographic noise almost entirely.


Mathematical modeling of erythrocyte chimerism informs genetic intervention strategies for sickle cell disease
Altrock, P.M., et al., 2016. Mathematical modeling of erythrocyte chimerism informs genetic intervention strategies for sickle cell disease. American Journal of Hematology , 91 (9) , pp. 931-937. Publisher's VersionAbstract

Recent advances in gene therapy and genome-engineering technologies offer the opportunity to correct sickle cell disease (SCD), a heritable disorder caused by a point mutation in the β-globin gene. The developmental switch from fetal γ-globin to adult β-globin is governed in part by the transcription factor (TF) BCL11A. This TF has been proposed as a therapeutic target for reactivation of γ-globin and concomitant reduction of β-sickle globin. In this and other approaches, genetic alteration of a portion of the hematopoietic stem cell (HSC) compartment leads to a mixture of sickling and corrected red blood cells (RBCs) in periphery. To reverse the sickling phenotype, a certain proportion of corrected RBCs is necessary; the degree of HSC alteration required to achieve a desired fraction of corrected RBCs remains unknown. To address this issue, we developed a mathematical model describing aging and survival of sickle-susceptible and normal RBCs; the former can have a selective survival advantage leading to their overrepresentation. We identified the level of bone marrow chimerism required for successful stem cell-based gene therapies in SCD. Our findings were further informed using an experimental mouse model, where we transplanted mixtures of Berkeley SCD and normal murine bone marrow cells to establish chimeric grafts in murine hosts. Our integrative theoretical and experimental approach identifies the target frequency of HSC alterations required for effective treatment of sickling syndromes in humans. Our work replaces episodic observations of such target frequencies with a mathematical modeling framework that covers a large and continuous spectrum of chimerism conditions.

The cancer stem cell fraction in hierarchically organized tumors can be estimated using mathematical modeling and patient-specific treatment trajectories
Werner, B., et al., 2016. The cancer stem cell fraction in hierarchically organized tumors can be estimated using mathematical modeling and patient-specific treatment trajectories. Cancer Research , 76 , pp. 1705-1713. Publisher's VersionAbstract

Many tumors are hierarchically organized and driven by a sub-population of tumor initiating cells (TICs), or cancer stem cells. TICs are uniquely capable of recapitulating the tumor and are implied to be highly resistant to radio- and chemotherapy. Macroscopic patterns of tumor expansion before treatment and tumor regression during treatment are tied to the dynamics of TICs. Until now, quantitative information about the fraction of TICs from macroscopic tumor burden trajectories could not be inferred. In this study, we generated a quantitative method based on a mathematical model that describes hierarchically organized tumor dynamics and patient-derived tumor burden information. The method identifies two characteristic equilibrium TIC regimes during expansion and regression. We show that tumor expansion and regression curves can be leveraged to infer estimates of the TIC fraction in individual patients at detection and after continued therapy. Furthermore, our method is parameter-free; it solely requires knowledge of a patient's tumor burden over multiple time points to reveal microscopic properties of the malignancy. We demonstrate proof of concept in the case of chronic myeloid leukemia (CML), wherein our model recapitulated the clinical history of the disease in two independent patient cohorts. Based on patient-specific treatment responses in CML, we predict that after one year of targeted treatment, the fraction of TICs increases 100-fold and continues to increase up to 1000-fold after five years of treatment. Our novel framework may significantly influence the implementation of personalized treatment strategies and has the potential for rapid translation into the clinic.

The mathematics of cancer: integrating quantitative models
Altrock, P.M., Liu, L. & Michor, F., 2015. The mathematics of cancer: integrating quantitative models. Nature Reviews Cancer , 15 , pp. 730-745. Publisher's VersionAbstract

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.

Which games are growing bacterial populations playing?
Li, X.-Y., et al., 2015. Which games are growing bacterial populations playing?. Journal of the Royal Society Interface , 12 , pp. 20150122. Publisher's VersionAbstract

Microbial communities display complex population dynamics, both in frequency and absolute density. Evolutionary game theory provides a natural approach to analyse and model this complexity by studying the detailed interactions among players, including competition and conflict, cooperation and coexistence. Classic evolutionary game theory models typically assume constant population size, which often does not hold for microbial populations. Here, we explicitly take into account population growth with frequency-dependent growth parameters, as observed in our experimental system. We study the in vitro population dynamics of the two commensal bacteria (Curvibacter sp. (AEP1.3) and Duganella sp. (C1.2)) that synergistically protect the metazoan hostHydra vulgaris (AEP) from fungal infection. The frequency-dependent, nonlinear growth rates observed in our experiments indicate that the interactions among bacteria in co-culture are beyond the simple case of direct competition or, equivalently, pairwise games. This is in agreement with the synergistic effect of anti-fungal activity observed in vivo. Our analysis provides new insight into the minimal degree of complexity needed to appropriately understand and predict coexistence or extinction events in this kind of microbial community dynamics. Our approach extends the understanding of microbial communities and points to novel experiments.

Complexity and stability in growing cancer cell populations
Gerlee, P. & Altrock, P.M., 2015. Complexity and stability in growing cancer cell populations. PNAS , 112 (21) , pp. E2742–E2743. Publisher's VersionAbstract

The study by Archetti et al. ( demonstrates frequency-dependent growth rates of two phenotypically distinct cancer subclones. One clone produced the insulin-like growth factor (IGF)-II, the other did not. In a mix of producers and nonproducers, the growth rates of both clones varied with the frequency of producers. Because a similar effect was shown when varying the concentration of serum, the production of IGF-II could be viewed as a public goods game. We welcome these experimental results but have serious concerns about the theoretical framework used for explaining them. Evolutionary game theory has certain advantages when it comes to understanding complex interactions, but further evidence is needed for its application to growing tumors. 

Fixation in finite populations evolving in fluctuating environments
Ashcroft, P., Altrock, P.M. & Galla, T., 2014. Fixation in finite populations evolving in fluctuating environments. Journal of the Royal Society Interface , 11 , pp. 20140663. Publisher's VersionAbstract

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. 

Non-cell-autonomous driving of tumourgrowth supports sub-clonal heterogeneity
Marusyk, A., et al., 2014. Non-cell-autonomous driving of tumourgrowth supports sub-clonal heterogeneity. Nature , 514 , pp. 54–58. Publisher's VersionAbstract

Cancers arise through a process of somatic evolution that can result in substantial sub-clonal heterogeneity within tumours. 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 tumour phenotypes and the competitive expansion ofindividual clones.We found that tumour growth can be driven by a minor cell subpopulation, which enhances the proliferation of all cells within a tumour by overcoming environmental constraints and yet can be outcompeted by faster proliferating competitors, resulting in tumour collapse. We developed a mathematical modelling framework to identify the rules underlying the generation of intra-tumour clonal heterogeneity. We found that non-cell-autonomous driving of tumour growth, together with clonal interference, stabilizes sub-clonal heterogeneity, thereby enabling inter-clonal interactions that can lead to new phenotypic traits.

First Steps towards Underdominant Genetic Transformation of Insect Populations
Reeves, R.G., et al., 2014. First Steps towards Underdominant Genetic Transformation of Insect Populations. PLoS One , 9 , pp. e97557. Publisher's VersionAbstract

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-GAL4driver 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.

Aspiration Dynamics of Multi-player Games in Finite Populations
Du, J., et al., 2014. Aspiration Dynamics of Multi-player Games in Finite Populations. Journal of the Royal Society Interface , 11 , pp. 20140077.Abstract
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 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 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.
The Evolution of Strategic Timing in Collective-Risk Dilemmas
Hilbe, C., et al., 2013. The Evolution of Strategic Timing in Collective-Risk Dilemmas. PLoS One , 8 , pp. e66490.Abstract
In collective-risk dilemmas, a group needs to collaborate over time to avoid a catastrophic event. This gives rise to a coordination game with many equilibria, including equilibria where no one contributes, and thus no measures against the catastrophe are taken. In this game, the timing of contributions becomes a strategic variable that allows individuals to interact and influence one another. Herein, we use evolutionary game theory to study the impact of strategic timing on equilibrium selection. Depending on the risk of catastrophe, we identify three characteristic regimes. For low risks, defection is the only equilibrium, whereas high risks promote equilibria with sufficient contributions. Intermediate risks pose the biggest challenge for cooperation. In this risk regime, the option to interact over time is critical; if individuals can contribute over several rounds, then the group has a higher chance to succeed, and the expected welfare increases. This positive effect of timing is of particular importance in larger groups, where successful coordination becomes increasingly difficult.
Bacterial colonization of Hydra hatchlings follows a robust temporal pattern
Franzenburg, S., et al., 2013. Bacterial colonization of Hydra hatchlings follows a robust temporal pattern. The ISME Journal , 7 , pp. 781-790.Abstract

Animals are colonized by complex bacterial communities. The processes controlling community membership and influencing the establishment of the microbial ecosystem during development are poorly understood. Here we aimed to explore the assembly of bacterial communities in Hydra with the broader goal of elucidating the general rules that determine the temporal progression of bacterial colonization of animal epithelia. We profiled the microbial communities in polyps at various time points after hatching in four replicates. The composition and temporal patterns of the bacterial communities were strikingly similar in all replicates. Distinct features included high diversity of community profiles in the first week, a remarkable but transient adult-like profile 2 weeks after hatching, followed by progressive emergence of a stable adult-like pattern characterized by low species diversity and the preponderance of the Betaproteobacterium Curvibacter. Intriguingly, this process displayed important parallels to the assembly of human fecal communities after birth. In addition, a mathematical modeling approach was used to uncover the organizational principles of this colonization process, suggesting that both, local environmental or host-derived factor(s) modulating the colonization rate, as well as frequency-dependent interactions of individual bacterial community members are important aspects in the emergence of a stable bacterial community at the end of development.

The mechanics of stochastic slowdown in evolutionary games
Altrock, P.M., Traulsen, A. & Galla, T., 2012. The mechanics of stochastic slowdown in evolutionary games. Journal of Theoretical Biology , 311 , pp. 94-106.Abstract
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 in the Prisoner’s Dilemma: Defection spreads fast initially, but the final steps to takeover can be delayed substantially. The analysis of coexistence games reveals even more intricate non-monotonic 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 generic 2 x 2 games with respect to the possibility to observe non-monotonic behavior of the conditional average fixation time as a function of selection strength.
Stability Properties of Underdominance in Finite Subdivided Populations
Altrock, P.M., Traulsen, A. & Reed, F.A., 2011. Stability Properties of Underdominance in Finite Subdivided Populations. PLoS Computational Biology , 7 , pp. e1002260. Publisher's VersionAbstract

Underdominance is a component of natural evolution: homozygotes – of either wildtypes or mutants – are advantageous. This can play a role in speciation and as a method to establish artificial genetic constructs in wild populations. The polymorphic state of wildtype and mutant alleles is unstable. However, in subdivided populations limited gene flow can counterbalance this effect. The maintenance of polymorphism sensitively depends on the amount of gene flow. In populations of finite size, the polymorphism is ultimately lost due to stochastic fluctuations, but there are long intermediate periods of polymorphism persistence. We analyze a simple population genetic model to characterize and explore the polymorphic phases depending on population size and genotypic fitness values. Even for large fluctuations (small population size), long periods of neither extinction nor fixation are possible. Since underdominance has been proposed as a genetic strategy in the pest management of disease vectors, it is important to understand the basic features of this system precisely, especially with a focus on gene flow between ecological patches. We assess different release strategies of potentially underdominant mutants, where one seeks to minimize the probability of fixation of the introduced allele but maximize the time to its extinction.

Stochastic slowdown in evolutionary processes
Altrock, P.M. & Traulsen, A., 2010. Stochastic slowdown in evolutionary processes. Physical Review E , 82  , pp. 011925. Publisher's VersionAbstract

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 nonvanishing 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.

Using underdominance to bi-stably transform local populations
Altrock, P.M., et al., 2010. Using underdominance to bi-stably transform local populations. Journal of Theoretical Biology , 267 , pp. 62-75.Abstract
Underdominance refers to natural selection against individuals with a heterozygous genotype. Here, we analyze 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.
Universality of weak selection
Wu, B., et al., 2010. Universality of weak selection. Physical Review E , 82 , pp. 046106.Abstract

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.

Deterministic evolutionary game dynamics in finite populations
Altrock, P.M. & Traulsen, A., 2009. Deterministic evolutionary game dynamics in finite populations. Physical Review E , 80 , pp. 011909.Abstract
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 present a 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 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 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 nontrivial way.