Journal of Mathematical Biology

, Volume 74, Issue 7, pp 1589–1609 | Cite as

Abrupt transitions to tumor extinction: a phenotypic quasispecies model

  • Josep Sardanyés
  • Regina Martínez
  • Carles Simó
  • Ricard Solé
Article

Abstract

The dynamics of heterogeneous tumor cell populations competing with healthy cells is an important topic in cancer research with deep implications in biomedicine. Multitude of theoretical and computational models have addressed this issue, especially focusing on the nature of the transitions governing tumor clearance as some relevant model parameters are tuned. In this contribution, we analyze a mathematical model of unstable tumor progression using the quasispecies framework. Our aim is to define a minimal model incorporating the dynamics of competition between healthy cells and a heterogeneous population of cancer cell phenotypes involving changes in replication-related genes (i.e., proto-oncogenes and tumor suppressor genes), in genes responsible for genomic stability, and in house-keeping genes. Such mutations or loss of genes result into different phenotypes with increased proliferation rates and/or increased genomic instabilities. Despite bifurcations in the classical deterministic quasispecies model are typically given by smooth, continuous shifts (i.e., transcritical bifurcations), we here identify a novel type of bifurcation causing an abrupt transition to tumor extinction. Such a bifurcation, named as trans-heteroclinic, is characterized by the exchange of stability between two distant fixed points (that do not collide) involving tumor persistence and tumor clearance. The increase of mutation and/or the decrease of the replication rate of tumor cells involves this catastrophic shift of tumor cell populations. The transient times near bifurcation thresholds are also characterized, showing a power law dependence of exponent \(-1\) of the transients as mutation is changed near the bifurcation value. These results are discussed in the context of targeted cancer therapy as a possible therapeutic strategy to force a catastrophic shift by simultaneously delivering mutagenic and cytotoxic drugs inside tumor cells.

Keywords

Applied mathematics Bifurcations Cancer evolution Dynamical systems Genomic instability Phenotypic model Quasispecies 

Mathematics Subject Classification

37N25 37G99 92B05 92D25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Josep Sardanyés
    • 1
    • 2
  • Regina Martínez
    • 3
  • Carles Simó
    • 4
  • Ricard Solé
    • 1
    • 2
    • 5
  1. 1.ICREA-Complex Systems Lab, Department of Experimental and Health SciencesUniversitat Pompeu FabraBarcelonaSpain
  2. 2.Institut de Biologia EvolutivaCSIC-Universitat Pompeu FabraBarcelonaSpain
  3. 3.Departament de Matemàtiques, Edifici C. Facultat de CiènciesUniversitat Autònoma de BarcelonaBarcelonaSpain
  4. 4.Departament de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain
  5. 5.The Santa Fe InstituteSanta FeUSA

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