Stochastic Multiscale Models of Cell Population Dynamics: Asymptotic and Numerical Methods

P. Guerrero^1, 2 ; T. Alarcón^2, 3

¹ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
² Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain
³ Departament de Matemàtiques, Universitat Atonòma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Mathematical modelling of natural phenomena, Tome 10 (2015) no. 1, pp. 64-93.

Résumé

In this paper we present a new methodology that allows us to formulate and analyse stochastic multiscale models of the dynamics of cell populations. In the spirit of existing hybrid multiscale models, we set up our model in a hierarchical way according to the characteristic time scales involved, where the stochastic population dynamics is governed by the birth and death rates as prescribed by the corresponding intracellular pathways (e.g. stochastic cell-cycle model). The feed-back loop is closed by the coupling between the dynamics of the population and the intracellular dynamics via the concentration of oxygen: Cells consume oxygen which, in turn, regulate the rate at which cells proceed through their cell-cycle. The coupling between intracellular and population dynamics is carried out through a novel method to obtain the birth rate from the stochastic cell-cycle model, based on a mean-first passage time approach. Cell proliferation is assumed to be activated when one or more of the proteins involved in the cell-cycle regulatory pathway hit a threshold. This view allows us to calculate the birth rate as a function of the age of the cell and the extracellular oxygen in terms of the corresponding mean-first passage time. We then proceed to formulate the stochastic dynamics of the population of cells in terms of an age-structured Master Equation. Further, we have developed generalisations of asymptotic (WKB) methods for our age-structured Master Equation as well as a τ −leap method to simulate the evolution of our age-structured population. Finally, we illustrate this general methodology with a particular example of a cell population where progression through the cell-cycle is regulated by the availability of oxygen.

Détail

DOI : 10.1051/mmnp/201510104

Bibliographie

[1] T. Alarcón, H.M. Byrne, P.K. Maini J. theor. Biol. 2004 395 411

[2] T. Alarcón, H.M. Byrne, P.K. Maini Multi-scale Model. Sim 2005 440 475

[3] T. Alarcon, K.M. Page J. R. Soc.Interface 2007 283 304

[4] B. Albert, D. Bray, J. Lewis, M. Raff, K. Roberts, J. D. Walter. Molecular biology of the cell. Garland Publishing, New York, 3rd edition, 1994.

[5] A.R.A. Anderson, M.A.J. Chaplain Bull. Math. Biol. 1998 857 899

[6] K. Ball, T.G. Kurtz, L. Popovic, G. Rempala Ann. App. Prob. 2006 1925 1961

[7] B. Bedessem, A. Stephanou Math. Biosci. 2014 31 39

[8] J.J. Binney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman. The theory of critical phenomena. Oxford University Press, Oxford, UK (1992).

[9] A.H. Box, D.J. Demetrick. Cell-cycle kinase inhibitor expression and hypoxia-induced cell-cycle arrest in human cell lines. Carcino-genesis, (2004), 2325–2335.

[10] C.J.W. Breward, H.M. Byrne, C.E. Lewis J. Math. Biol. 2002 125 152

[11] H.M. Byrne Nat. Rev. Cancer 2010 221 230

[12] Y. Cao, D.T. Gillespie, L.R. Petzold J. Chem. Phys. 2006 1 11

[13] Y. Chen, R. Cairns, I. Papandreou, A. Koong, N.C. Denko PLoS One 2009

[14] T.S. Deisboeck, Z. Wang, P. Macklin, V. Cristini Annu. Rev. Biomed. Eng. 2011 127 155

[15] R. Durrett Notices of the AMS. 2013 304 309

[16] J.O. Funk Anticancer Res. 1999 4778 4780

[17] C.W. Gardiner. Stochastic methods. Springer-Verlag, Berlin, Germany (2009).

[18] L.B. Gardner, Q. Li, M.S. Parks, W.M. Flanagan, G.L. Semanza, C.V. Dang J. Biol. Chem. 2001 77919 7926

[19] M.A. Gibson, J. Bruck J. Phys. Chem. A. 2000 1876 1889

[20] D.T. Gillespie J. Comp. Phys. 1976 403 434

[21] D.T. Gillespie J. Chem. Phys. 2001 1716 1733

[22] D.T. Gillespie, L.R. Petzold J. Chem. Phys. 2003 8229 8234

[23] S.L. Green, R.A. Freiberg, A.J. Giaccia Mol. Cell Biol. 2001 1196 1206

[24] G. Grimmet, D. Stirzaker. Probability and random processes. Oxford University Press, Oxford, UK (2001)

[25] P.E. Hand, B.E. Griffith Proc. Natl. Acad. Sci. 2010 14603 14608

[26] S.D. Hester, J.M. Belmonte, J.S. Gens, S.G. Clendenon, J.A. Glazier PLoS Comput. Biol. 2011

[27] Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J.P. Freyer Biophys. J. 2005 38843894

[28] Y.J. Kim, H.G. Othmer Bull. Math. Biol. 2013 1304 1350

[29] H. Kitano Nat. Rev. Cancer. 2004 227 235

[30] R. Kubo, K. Matsuo, K. Kitahara J. Stat. Phys. 1973 51 96

[31] S. Land, S.A. Niederer, W.E. Louch, O.M. Sejersted, N.P. Smith J. R. Soc. Interface Focus. 2013 20120076

[32] F. Lignet, V. Calvez, E. Grenier, B. Ribba Math. Biosci. Eng. 2013 167 84

[33] J.S. Lowengrub, H.B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S.M. Wise, V. Cristini. Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity. 23:R1R91, 2010.

[34] P. Macklin, S. Mcdougall, A.R.A. Anderson, M.A.J. Chaplain, V. Cristini, J. Lowengrub J. Math. Biol. 2009 765 798

[35] R.S. Maier, D.L. Stein J. Stat. Phys. 1996 291 357

[36] A.D. McCulloch. Systems biology and multiscale modelling of the heart. In Proccedings of the Biomedical Science Engineering Conference, 2009. BSEC 2009. First Annual ORNL., 1-3, 2009.

[37] S.R. Mcdougall, A.R.A. Anderson, M.A.J. Chaplain, J.A. Sherratt Bull. Math. Biol. 2002 673 702

[38] S.R. Mcdougall, A.R.A. Anderson, M.A.J. Chaplain J. Theor. Biol. 2006 564 589

[39] J.A. Menendez, J. Joven, S. Cufi, B. Corominas-Faja, C. Oliveras-Ferraros, E. Cuyas, B. Martin-Castillo, E. Lopez-Bonet, T. Alarcon, A. Vazquez-Martin Cell Cycle 2013 1166 1179

[40] A.C. Oates, N. Gorfinkiel, M. Gonzalez-Gaitan, C.-P. Heisenberg Nature Rev. Gen. 2009 517 530

[41] J.M. Osborne, A. Walter, S.K. Kershaw, G.R. Mirams, A.G. Fletcher, P. Pathmanathan, D. Gavaghan, O.E. Jensen, P.K. Maini, H.M. Byrne Phil. Trans. R. Soc., A. 2010 5013 5028

[42] M.R. Owen, T. Alarcón, H.M. Byrne, P.K. Maini J. Math. Biol. 2009 689 721

[43] H. Perfahl, H.M. Byrne, T. Chen, V. Estrella, T. Alarcon, A. Lapin, R.A. Gatenby, R.J. Gillies, M.C. Lloyd, P.K. Maini, M. Reuss, M.R. Owen PLoS One. 2011

[44] G.G. Powathil, D.J.A. Adamson, M.A.J. Chaplain PLoS Comput. Biol. 2013

[45] L. Preziosi, A. Tosin Math. Model. Nat. Phenom. 2009 1 11

[46] S. Redner. A guide to first-passage processes. Cambridge University Press, Cambridge, UK, 2001.

[47] K.A. Rejniak, A.R.A. Anderson. Multi-scale hybrid models of tumor growth. Wiley Interdisciplinary Reviews: System Biology and Medicine. 2010.

[48] B. Ribba, T. Collin, S. Schnell Theor. Biol. Med. Model. 2006

[49] S. Schnell, P.K. Maini, S.A. Newman, T.J. Newman (eds.). Multi-scale Modeling of Developmental Systems. Academic Press, Oxford, UK, 2008.

[50] A. Stephanou, S.R. Mcdougall, A.R.A. Anderson, M.A.J. Chaplain Math. Comp. Model. 2005 1137 1156

[51] Y. Setty Bioinformatics 2012 20222028

[52] N.P. Smith, D.P. Nickerson, E.J. Crampin, P.J. Hunter Acta Numerica. 2004 371 431

[53] K. Strebhardt, A. Ullrich Nat. Rev. Cancer 2008 473 480

[54] D.W. Stacey Current Opinion in Cell Biology 2003 158 163

[55] A. Szabo, R.M.H. Merks Front. Oncol. 2013 87

[56] P. Tracqui Rep. Progr. Phys. 2009 056701

[57] R.D.M. Travasso, E. Corvera Poiré, M. Castro, J.C. Rodrguez-Manzaneque, A. Hernndez-Machado. Tumor angiogenesis and vascular patterning: A mathematical model. PLoS One. 6: e19989, 2011.

[58] J.J. Tyson, B. Novak J. theor. Biol. 2001 249 263

[59] N.G. Van Kampen. Stochastic processes in Physics and Chemistry. Elsevier, The Netherlands, (2007).

[60] F. Van Drogen, O. Sangfelt, A. Malyukova, L. Matskova, E. Yeh, A.R. Means, S.I. Reed Mol. Cell. 2006 37 48

[61] R. Visintin, S. Prinz, A. Amon Science 1997 460 463

[62] J. Walpole, J.A. Papin, S.M. Peirce Annu. Rev. Biomed. Eng. 2013 137 154

[63] W. Zachariae, K. Nasmyth Genes Devel. 1999 2039 2058