Advertisement

Deterministic and Stochastic Dynamics of Chronic Myelogenous Leukaemia Stem Cells Subject to Hill-Function-Like Signaling

  • Tor Flå
  • Florian Rupp
  • Clemens WoywodEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 35)

Abstract

Based on a discrete Markovian birth-death model including regulated symmetric and asymmetric cell division, we formulate a continuous four-dimensional stochastic (ordinary) differential equation model for the dynamics of Chronic Myelogenous Leukaemia (CML) stem cells in a bone marrow niche involving signaling and competition between active stem cells. Invoking stochastic-deterministic correspondence we then investigate two deterministic subsystems: (a) the competition between active normal and wild-type CML stem cells or also between two developing leukaemic stem cell strains is represented by a two-dimensional equation system, and (b) a three-dimensional model involving both cycling and noncycling normal stem cells as well as cycling wild-type CML stem cells is defined. The four-dimensional equation system finally includes in addition one cycling CML stem cell clone of an anti-CML-drug-resistant mutant. By totally analytic means we discuss the existence and stability of the equilibria of the three systems in the deterministic small noise limit, and establish, by numerical means, connections between these classical results and the original stochastic setting. The robust, stable finite population equilibria can be interpreted as homeostatic equilibria of normal and leukaemic stem cell populations, in the case of the four-dimensional model for the scenario of treatment of the wild-type CML clone with a CML suppressing agent, e.g., imatinib, which leads to the emergence of a resistant CML strain. The four-dimensional model thus represents a common clinical picture.

Keywords

Stem Cell Chronic Myelogenous Leukaemia Stem Cell Population Stem Cell Niche Normal Stem Cell 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

C.W. would like to thank the Mohn Foundation, the Centre for Theoretical and Computational Chemistry (CTCC) at the University of Tromsø and the Research Council of Norway (Grant Nr. 177558/V30) for continued support.

We also acknowledge computational resources provided by Norwegian High Performance Computing (NOTUR).

Special thanks go to Jürgen Scheurle, who brought F.R. back to academia.

References

  1. 1.
    Ackleh, A.S., Hu, S.: Comparison between stochastic and deterministic selection-mutation models. Math. Biosci. Eng. 4, 133–157 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ackleh, A.S., Hu, S.: Global dynamics of hematopoietic stem cells and differentiated cells in a chronic myeloid leukemia model. J. Math. Biol. 62, 975–997 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alon, U.: An Introduction to Systems Biology—Design Principles of Biological Circuits. Mathematical and Computational Biology. Chapman and Hall/CRC/Taylor and Francis Group, Boca Raton (2007)Google Scholar
  4. 4.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (2003)Google Scholar
  5. 5.
    Bai, F., Wu, Z., Jin, J., Hochendoner, P., Xing, J.: Slow protein conformational change, allostery and network dynamics. In: Cai, W., Hong, H. (eds.) Protein-Protein Interactions - Computational and Experimental Tools. InTech, Shanghai (2012)Google Scholar
  6. 6.
    Buchdunger, E., Zimmermann, J., Mett, H., Meyer, T., Müller, M., Druker, B.J., Lydon, N.B.: Inhibition of the abl protein-tyrosine kinase in vitro and in vivo by a 2-phenylaminopyrimidine derivative. Cancer Res. 56, 100 (1996)Google Scholar
  7. 7.
    Devys, A., Goudon, T., Lafitte, P.: A model describing the growth and the size distribution of multiple metastatic tumors. Discrete Contin. Dyn. B 12, 731–767 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Doumic-Jauffet, M., Kim, P.S., Perthame, B.: Stability analysis of a simplified yet complete model for chronic myelogenous leukemia. Bull. Math. Biol. 72, 1732–1759 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Flå, T., Ahmed, S.H.: Evolution of cold adapted protein sequences. In: Fung, G. (ed.) Sequence and Genome Analysis: Methods and Application. II. iConcept Press Ltd, Brisbane (2011)Google Scholar
  10. 10.
    Foo, J., Drummond, M.W., Clarkson, B., Holyoake, T., Michor, F.: Eradication of chronic myeloid leukemia stem cells: a novel mathematical model predicts no therapeutic benefit of adding g-csf to imatinib. PLoS Comput. Biol. 5(9), e1000503 (2009). URL http://view.ncbi.nlm.nih.gov/pubmed/19749982
  11. 11.
    Foo, J., Michor, F.: Evolution of resistance to anti-cancer therapy during general dosing schedules. J. Theor. Biol. 263(2), 179–88 (2010). URL http://view.ncbi.nlm.nih.gov/pubmed/20004211 Google Scholar
  12. 12.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  13. 13.
    Friedman, A.: A hierarchy of cancer models and their mathematical challenges. Discrete Contin. Dyn. B 4, 147–159 (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    Friedman, A.: Stochastic Differential Equations and Applications. Dover Publications, New York (2006)zbMATHGoogle Scholar
  15. 15.
    Friedman, A., Kim, Y.: Tumor cells proliferation and migration under the influence of their microenvironment. Math. Biosci. Eng. 8, 371–383 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fujarewicz, K., Kimmel, M., Swierniak, A.: On fitting of mathematical models of cell signaling pathways using adjoint systems. Math. Biosci. Eng. 2, 527–534 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gilsing, H., Shardlow, T.: Sdelab: A package for solving stochastic differential equations in matlab. J. Comput. Appl. Math. 205, 1002–1018 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Glauche, I., Lorenz, R., Hasenclever, D., Roeder, I.: A novel view on stem cell development: analysing the shape of cellular genealogies. Cell Prolif. 42(2), 248–263 (2009). URL http://view.ncbi.nlm.nih.gov/pubmed/19254328 Google Scholar
  19. 19.
    Glauche, I., Moore, K., Thielecke, L., Horn, K., Loeffler, M., Roeder, I.: Stem cell proliferation and quiescence—two sides of the same coin. PLoS Comput. Biol. 5(7), e1000447 (2009). URL http://view.ncbi.nlm.nih.gov/pubmed/19629161
  20. 20.
    Gruber, F.X.: Towards a quantitative understanding of cml resistance. Ph.D. Thesis. Department of Pharmacology, University of Tromsø (2009)Google Scholar
  21. 21.
    Gruber, F.X., Ernst, T., Porkka, K., Engh, R., Mikkola, I., Maier, J., Lange, T., Hochhaus, A.: Dynamics of the emergence of dasatinib and nilotinib resistance in imatinib resistant cml patients. Leukemia 26, 172 (2012). http://dx.doi.org/10.1038/leu.2011.187. URL http://www.nature.com/leu/journal/v26/n1/full/leu2011187a.html
  22. 22.
    Hlavacek, W.S., Faeder, J.R.: The complexity of cell signaling and the need for a new mechanics. Sci. Signal. 2(81), pe46 (2009). DOI 10.1126/ scisignal.281pe46. URL http://stke.sciencemag.org/cgi/content/abstract/sigtrans;2/81/pe46
  23. 23.
    Horsthemke, W., Lefever, R.: Noise-Induced Transitions – Theory ans Applications in Physics, Chemistry, and Biology. Springer, Berlin (1984)Google Scholar
  24. 24.
    Kimura, M.: Diffusion models in population genetics. J. Appl. Probab. 1(2), 177–232 (1964). URL http://www.jstor.org/stable/3211856 Google Scholar
  25. 25.
    Kimura, M., Crow, J.F.: The measurement of effective population number. Evolution 17(3), 279–288 (1963). URL http://www.jstor.org/stable/2406157 Google Scholar
  26. 26.
    Kloeden, P.E.: The systematic derivation of higher order numerical methods for stochastic differential equations. Milan J. Math. 70, 187–207 (2002)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kloeden, P.E., Neukirch, A.: Convergence of numerical methods for stochastic differential equations in mathematical finance (2012). arXiv:1204.6620v1 [math.NA]Google Scholar
  28. 28.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999)Google Scholar
  29. 29.
    Kloeden, P.E., Platen, E., Wright, I.W.: The approximation of multiple stochastic integrals. Stoch. Anal. Appl. 10, 431–441 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Komarova, N.L.: Mathematical modeling of cyclic treatment of chronic myeloid leukemia. Math. Biosci. Eng. 8, 289–306 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Komarova, N.L., Katouli, A.A., Wodarz, D.: Combination of two but not three current targeted drugs can improve therapy of chronic myeloid leukemia. PLoS One 4(2), e4423 (2009). URL http://view.ncbi.nlm.nih.gov/pubmed/19204794
  32. 32.
    Komarova, N.L., Wodarz, D.: Drug resistance in cancer: principles of emergence and prevention. Proc. Natl. Acad. Sci. USA 102, 9714 (2005)CrossRefGoogle Scholar
  33. 33.
    Komarova, N.L., Wodarz, D.: Effect of cellular quiescence on the success of targeted cml therapy. PLoS One 2(10), e990 (2007). URL http://view.ncbi.nlm.nih.gov/pubmed/17912367
  34. 34.
    Komarova, N.L., Wodarz, D.: Stochastic modeling of cellular colonies with quiescence: an application to drug resistance in cancer. Theor. Popul. Biol. 72(4), 523–538 (2007). URL http://view.ncbi.nlm.nih.gov/pubmed/17915274 Google Scholar
  35. 35.
    Komarova, N.L., Wodarz, D.: Combination therapies against chronic myeloid leukemia: short-term versus long-term strategies. Cancer Res. 69(11), 4904–4910 (2009). URL http://view.ncbi.nlm.nih.gov/pubmed/19458080 Google Scholar
  36. 36.
    Li, W., Wolynes, P.G., Takada, S.: Frustration, specific sequence dependence, and nonlinearity in large-amplitude fluctuations of allosteric proteins. Proc. Natl. Acad. Sci. 108(9), 3504–3509 (2011). DOI 10.1073/pnas.1018983108. URL http://www.pnas.org/content/108/9/3504.abstract
  37. 37.
    Lin, Y.T., Kim, H., Doering, C.R.: Features of fast living: on the weak selection for longevity in degenerate birth-death processes. J. Stat. Phys. 148, 646–662 (2012)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Michor, F., Hughes, T.P., Iwasa, Y., Branford, S., Shah, N.P., Sawyers, C.L., Nowak, M.A.: Dynamics of chronic myeloid leukaemia. Nature 435(7046), 1267–1270 (2005). URL http://view.ncbi.nlm.nih.gov/pubmed/15988530 Google Scholar
  39. 39.
    Mjolsness, E.: On cooperative quasi-equilibrium models of transcriptional regulation. J. Bioinform. Comput. Biol. 5(2b), 467–490 (2007)CrossRefGoogle Scholar
  40. 40.
    Mjolsness, E.: Towards a calculus of biomolecular complexes at equilibrium. Brief. Bioinform. 8(4), 226–233 (2007)CrossRefGoogle Scholar
  41. 41.
    Paquin, D., Kim, P.S., Lee, P.P., Levy, D.: Strategic treatment interruptions during imatinib treatment of chronic myelogenous leukemia. Bull. Math. Biol. 73, 1082 (2010). URL http://view.ncbi.nlm.nih.gov/pubmed/20532990 Google Scholar
  42. 42.
    Roeder, I., Herberg, M., Horn, M.: An age-structured model of hematopoietic stem cell organization with application to chronic myeloid leukemia. Bull. Math. Biol. 71(3), 602–626 (2009). URL http://view.ncbi.nlm.nih.gov/pubmed/19101772 Google Scholar
  43. 43.
    Roeder, I., Horn, M., Glauche, I., Hochhaus, A., Mueller, M.C., Loeffler, M.: Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat. Med. 12(10), 1181–4 (2006). URL http://view.ncbi.nlm.nih.gov/pubmed/17013383 Google Scholar
  44. 44.
    Strook, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (2006)Google Scholar
  45. 45.
    Tomasetti, C., Levy, D.: Role of symmetric and asymmetric division of stem cells in developing drug resistance. Proc. Natl. Acad. Sci. USA 107, 16766–16771 (2010)CrossRefGoogle Scholar
  46. 46.
    Vallee-Belisle, A., Ricci, F., Plaxco, K.W.: Thermodynamic basis for the optimization of binding-induced biomolecular switches and structure-switching biosensors. Proc. Natl. Acad. Sci. 106(33), 13802–13807 (2009). DOI 10.1073/pnas.0904005106. URL http://www.pnas.org/content/106/33/13802.abstract
  47. 47.
    Walczak, A.M., Tkačik, G.c.v., Bialek, W.: Optimizing information flow in small genetic networks. ii. feed-forward interactions. Phys. Rev. E 81, 041905 (2010). DOI 10.1103/PhysRevE.81.041905. URL http://link.aps.org/doi/10.1103/PhysRevE.81.041905
  48. 48.
    Wang, G., Zaman, M.H.: Communications: Hamiltonian regulated cell signaling network. J. Chem. Phys. 132(12), 121103 (2010) DOI 10.1063/1.3357980. URL http://link.aip.org/link/?JCP/132/121103/1 Google Scholar
  49. 49.
    Wang, J., Huang, B., Xia, X., Sun, Z.: Funneled landscape leads to robustness of cell networks: yeast cell cycle. PLoS Comput. Biol. 2(11), e147 (2006). DOI 10.1371/journal.pcbi.0020147. URL http://dx.plos.org/10.1371%2Fjournal.pcbi.0020147
  50. 50.
    Waxman, D.: A unified treatment of the probability of fixation when population size and the strength of selection change over time. Genetics 188(4), 907–913 (2011). DOI 10.1534/genetics.111.129288. URL http://www.genetics.org/content/188/4/907.abstract
  51. 51.
    Wiktorsson, M.: Joint characteristic function and simultaneous simulation of iterated Ito integrals for multiple independent Brownian motions. Ann. Appl. Probab. 11, 470–487 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wodarz, D.: Stem cell regulation and the development of blast crisis in chronic myeloid leukemia: implications for the outcome of imatinib treatment and discontinuation. Med. Hypotheses 70, 128 (2008)CrossRefGoogle Scholar
  53. 53.
    Woywod, C., Gruber, F., Engh, R., Flå, T.: Dynamical models of mutated chronic myelogenous leukaemia cells for a post-imatinib treatment scenario: response to dasatinib or nilotinib therapy. PLoS Comput. Biol. (2013, submitted)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TromsøTromsøNorway
  2. 2.Lehrstuhl für Höhere Mathematik und Analytische MechanikTechnische Universität München, Fakultät für MathematikGarchingGermany
  3. 3.Centre for Theoretical and Computational Chemistry (CTCC), Department of ChemistryUniversity of TromsøTromsøNorway

Personalised recommendations