The publications on this website have not been updated since April 2018. For more recent contributions from our lab, please visit our new website

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Recent Publications

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. (http://www.pnas.org/content/112/6/1833) 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.

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