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Dynamics and bifurcations in a simple quasispecies model of tumorigenesis

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Abstract

Cancer is a complex disease and thus is complicated to model. However, simple models that describe the main processes involved in tumoral dynamics, e.g., competition and mutation, can give us clues about cancer behavior, at least qualitatively, also allowing us to make predictions. Here, we analyze a simplified quasispecies mathematical model given by differential equations describing the time behavior of tumor cell populations with different levels of genomic instability. We find the equilibrium points, also characterizing their stability and bifurcations focusing on replication and mutation rates. We identify a transcritical bifurcation at increasing mutation rates of the tumor cells. Such a bifurcation involves a scenario with dominance of healthy cells and impairment of tumor populations. Finally, we characterize the transient times for this scenario, showing that a slight increase beyond the critical mutation rate may be enough to have a fast response towards the desired state (i.e., low tumor populations) by applying directed mutagenic therapies.

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Notes

  1. The codes used to obtain the results presented in this work are available upon request.

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Acknowledgments

We want to thank Ricard V. Solé for helpful comments and suggestions. JTL has been partially supported by the Spanish MICIN/FEDER grant MTM2012-31714, by the Generalitat de Catalunya grant number 2014SGR-504, and by grant 14-41-00044 of RSF at the Lobachevsky University of Nizhny Novgorod. JS has been funded by the Fundación Botín.

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Authors and Affiliations

  1. Department of Applied Mathematics, Universitat Politècnica de Catalunya, Avinguda Diagonal 647, 08028, Barcelona, Spain

    Vanessa Castillo & J. Tomás Lázaro

  2. ICREA-Complex Systems Lab, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Parc de Recerca Biomèdica de Barcelona (PRBB), Dr. Aiguader 88, 08003, Barcelona, Spain

    Josep Sardanyés

  3. Institut de Biologia Evolutiva (CSIC-Universitat Pompeu Fabra), Passeig Maritim de la Barceloneta 37, 08003, Barcelona, Spain

    Josep Sardanyés

Authors
  1. Vanessa Castillo
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  2. J. Tomás Lázaro
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  3. Josep Sardanyés
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Corresponding authors

Correspondence to J. Tomás Lázaro or Josep Sardanyés.

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Communicated by Maria do Rosario de Pinho.

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Castillo, V., Lázaro, J.T. & Sardanyés, J. Dynamics and bifurcations in a simple quasispecies model of tumorigenesis. Comp. Appl. Math. 36, 415–431 (2017). https://doi.org/10.1007/s40314-015-0234-3

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