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Ricardo Borges, Àngel Calsina, Sílvia Cuadrado Universitat Autònoma de Barcelona

On a cyclin content structured cell population model. Ricardo Borges, Àngel Calsina, Sílvia Cuadrado Universitat Autònoma de Barcelona. Outline. Introduction The model Stationary solutions Asymtoptic behavior: x -independent solutions Numerical simulations.

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Ricardo Borges, Àngel Calsina, Sílvia Cuadrado Universitat Autònoma de Barcelona

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  1. On a cyclin content structured cell population model Ricardo Borges, Àngel Calsina, Sílvia Cuadrado Universitat Autònoma de Barcelona

  2. Outline • Introduction • The model • Stationary solutions • Asymtoptic behavior: • x-independent solutions • Numerical simulations

  3. M. Gyllenberg, G. Webb Age-size structure in populations with quiescence Math. Biosc. (1987) “it is hypothesized that growth arrest states in the G1 phase of the cell cycle are closely related to an integrated control of cell prolliferation and differentiation, and that cancer may result form defects that uncouple this integrated system” M. Gyllenberg, G. Webb A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol. (1990) “A realistic description of proliferation and quiescence in tumors require a structured model”

  4. In 2001, Leland H. Hartwell, R. Timothy Hunt, and Paul M. Nurse won the Nobel Prize in Medicine and Physiology for their discoveries regarding cell cycle regulation by cyclin and cyclin-dependent kinases. • F.B. Bricki, J. Clairambault, B. Ribba, B. Perthame, An age-and-cyclin-structured population model for healthy and tumoral tissues. J. Math. Biol. (2008) • “Unlimited tumor growth can be seen in particular as a deregulation of transitions between proliferative and quiescent compartments. Furthermore, recent measurements indicate that cyclins are the most determinant control molecules for phase transitions” • M. Doumic, Analysis of a Population Model Structured by the cells molecular content. MMNP (2007) • R. Borges, A. C., S. Cuadrado, Equilibria of a cyclin structured cell population model. To appear in DCDS

  5. Cell cycle

  6. The model cyclin regulation of the transition to G0 phase G1-S-G2-M apoptosis apoptosis P(x, t) Q(x, t) cell division

  7. evolution speed of cyclin content in a particular cell

  8. cell division rate of cells with x cyclin content production of cells with x cyclin content per unit time:

  9. Density dependent recruitment function Cyclin dependent “demobilisation” function

  10. The model:

  11. Find in the range of function G and such that Steady states: P(x)and Q(x)such that • In abstract form: and

  12. such that Steady states: Find in the range of G and Under good hypothesis is a simple dominant eigenvalue with a corresponding positive eigenvector and it is the only eigenvalue with positive eigenvector. Moreover it is an increasing function of g.

  13. Let us further assume amounts to population decrease when there is no recruitment from the Q-stage and amounts to population increase when the population number is small and healthy tissue!

  14. Then there is a unique (nontrivial) steady state given by whereP^ is the (positive) eigenvector corresponding to the 0 eigenvalue of the operator normalized in such a way that

  15. Asymptotic behavior A particular case with x-independent solutions: normalize xmto 1 and take has solutions which satisfy the o.d.e. system

  16. Phase portrait for the x-independent solutions extinction

  17. Hopf bifurcation Phase portrait for the x-independent solutions a unique nontrivial steady state stable, but….

  18. Phase portrait for the x-independent solutions unbounded solutions

  19. Numerical simulations in the general caseT. Kostova, An explicit third-order numerical method for size-structured population equations, Num. Methods in PDE (2003) we have non local terms and moreover, the principal part Pt+(ΓP)x is such that Γ(x) vanishes at the ends of the domain.

  20. An explicit numerical method based on integrationalong characteristics lines. A non uniform rectangular grid with constant time step.(xi,tj) and (xi+1,tj+1) locatedon the same characteristic curve:

  21. Numerical simulations in the general case Extinction: extinction.gif

  22. Numerical simulations in the general case stabilization:

  23. Numerical simulations in the general case Oscillations:

  24. Numerical simulations in the general case • Unbounded solutions: UNBOUNDED.gif

  25. Conclusions • A simplification of a pre-existing model of two stages cell tissue growth has been considered • An analytical proof of existence and uniqueness of steady state for suitable density dependent transition rate from quiescent to proliferating stage • Convergence to the steady state; but also selfsustained oscillations of the populations (even in the case of healthy tissue) if the transition rate is large and very sensitive to changes in the populations • Numerical simulations corroborating the results

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