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.
Applied mathematics Bifurcations Cancer Complex systems Quasispecies dynamics
Mathematics Subject Classification
92Bxx Mathematical biology in general 65Lxx ODEs 58Kxx catastrophe theory 37G35 Attractors and their bifurcations 70K50 bifurcations and instability
This is a preview of subscription content, log in to check access.
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.
Anderson GR, Stoler DL, Brenner BM (2001) Cancer as an evolutionary consequence of a destabilized genome. Bioessays 23:103746CrossRefGoogle Scholar
Benedikt B, Siebert R, Traulsen A (2014) Cancer initiation with epistatic interaction between driver and passenger mutations. J Theor Biol 358:52–60CrossRefGoogle Scholar
Bielas JH, Loeb KR, Rubin BP, True LD et al (2006) Human cancers express a mutator phenotype. Proc Natl Acad Sci 103:18238–18242CrossRefGoogle Scholar
Bjedov I, Tenaillon O, Gérard B, Souza V et al (2003) Stress-induced mutagenesis in bacteria. Science 300(14049):35Google Scholar