2015, 12(3): 491-501. doi: 10.3934/mbe.2015.12.491

A model for asymmetrical cell division

1. 

Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand, New Zealand

2. 

Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

Received  August 2014 Revised  November 2014 Published  January 2015

We present a model that describes the growth, division and death of a cell population structured by size. The model is an extension of that studied by Hall and Wake (1989) and incorporates the asymmetric division of cells. We consider the case of binary asymmetrical splitting in which a cell of size $\xi$ divides into two daughter cells of different sizes and find the steady size distribution (SSD) solution to the non-local differential equation. We then discuss the shape of the SSD solution. The existence of higher eigenfunctions is also discussed.
Citation: Ali Ashher Zaidi, Bruce Van Brunt, Graeme Charles Wake. A model for asymmetrical cell division. Mathematical Biosciences & Engineering, 2015, 12 (3) : 491-501. doi: 10.3934/mbe.2015.12.491
References:
[1]

B. Basse, B. Baguley, E. Marshell, W. Joseph, B. Van-Brunt, G. C. Wake and D. Wall, Modelling cell death in human tumor cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004), 329-357. doi: 10.1007/s00285-003-0254-2.

[2]

Basse, G. C. Wake, D. J. N. Wall and B. Van-Brunt, On a cell-growth model for plankton, Mathematical medicine and biology, 21 (2004), 49-61.

[3]

R. Begg, Cell-population Growth Modeling and Functional Differential Equations, Ph.D thesis, University of Canterbury, New Zealand, 2007.

[4]

R. Begg, D. J. N. Wall and G. C. Wake, On a functional equation model of transient cell growth, Mathematical medicine and biology, 22 (2005), 371-390. doi: 10.1093/imammb/dqi015.

[5]

M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to self similarity for the fragmentation equation in $L^1$ spaces, Communications in Applied and Industrial Mathematics, 1 (2010), 299-308.

[6]

M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathémathiques Pures et Appliquée, 96 (2011), 334-362. doi: 10.1016/j.matpur.2011.01.003.

[7]

F. P. Da Costa, M. Grinfeld and J. B. Mcleod, Unimodality of steady size distributions of growing cell populations, J.evol.equ., 1 (2001), 405-409. doi: 10.1007/PL00001379.

[8]

O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, Jour. Math. Biol., 19 (1984), 227-248. doi: 10.1007/BF00277748.

[9]

A. J. Hall and G. C. Wake, A functional differential equation arising in modelling of cell growth, J. Aust. Math. Soc. Ser. B, 30 (1989), 424-435. doi: 10.1017/S0334270000006366.

[10]

A. J. Hall, G. C. Wake and P. W. Gandar, Steady size distributions for cells in one dimensional plant tissues, J. Math. Biol., 30 (1991), 101-123. doi: 10.1007/BF00160330.

[11]

H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Mathematical Biosciences, 72 (1984), 19-50. doi: 10.1016/0025-5564(84)90059-2.

[12]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510. doi: 10.4310/CMS.2009.v7.n2.a12.

[13]

T. R. Malthus, An Essay on the Principle of Population, St. Paul's London, 1798.

[14]

A. G. Mckendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130.

[15]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[16]

P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697-702. doi: 10.1016/j.crma.2004.03.006.

[17]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001.

[18]

R. A. Neumïler and J. A. Knoblich, Dividing cellular asymmetry: Asymmetric cell division and its implications for stem cells and cancer, Genes Dev., 23 (2009), 2675-2699.

[19]

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, Journal of Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018.

[20]

T. Suebcharoen, B. Van-Brunt and G. C. Wake, Asymmetric cell division in a size-structured growth model, Differential and Integral Equations, 24 (2011), 787-799.

[21]

B. Van-Brunt, G. C. Wake and H. K. Kim, A singular Sturm-Liouville problem involving an advanced functional differential equation, European Journal of Applied Mathematics, 12 (2001), 625-644. doi: 10.1017/S0956792501004624.

[22]

B. Van-Brunt and M. Vlieg-Hulstman, An eigenvalue problem involving a functional differential equation arising in a cell growth model, ANZIAM J., 51 (2010), 383-393. doi: 10.1017/S1446181110000866.

show all references

References:
[1]

B. Basse, B. Baguley, E. Marshell, W. Joseph, B. Van-Brunt, G. C. Wake and D. Wall, Modelling cell death in human tumor cell lines exposed to the anticancer drug paclitaxel, J. Math. Biol., 49 (2004), 329-357. doi: 10.1007/s00285-003-0254-2.

[2]

Basse, G. C. Wake, D. J. N. Wall and B. Van-Brunt, On a cell-growth model for plankton, Mathematical medicine and biology, 21 (2004), 49-61.

[3]

R. Begg, Cell-population Growth Modeling and Functional Differential Equations, Ph.D thesis, University of Canterbury, New Zealand, 2007.

[4]

R. Begg, D. J. N. Wall and G. C. Wake, On a functional equation model of transient cell growth, Mathematical medicine and biology, 22 (2005), 371-390. doi: 10.1093/imammb/dqi015.

[5]

M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to self similarity for the fragmentation equation in $L^1$ spaces, Communications in Applied and Industrial Mathematics, 1 (2010), 299-308.

[6]

M. J. Cáceres, J. A. Cañizo and S. Mischlerl, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathémathiques Pures et Appliquée, 96 (2011), 334-362. doi: 10.1016/j.matpur.2011.01.003.

[7]

F. P. Da Costa, M. Grinfeld and J. B. Mcleod, Unimodality of steady size distributions of growing cell populations, J.evol.equ., 1 (2001), 405-409. doi: 10.1007/PL00001379.

[8]

O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, Jour. Math. Biol., 19 (1984), 227-248. doi: 10.1007/BF00277748.

[9]

A. J. Hall and G. C. Wake, A functional differential equation arising in modelling of cell growth, J. Aust. Math. Soc. Ser. B, 30 (1989), 424-435. doi: 10.1017/S0334270000006366.

[10]

A. J. Hall, G. C. Wake and P. W. Gandar, Steady size distributions for cells in one dimensional plant tissues, J. Math. Biol., 30 (1991), 101-123. doi: 10.1007/BF00160330.

[11]

H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Mathematical Biosciences, 72 (1984), 19-50. doi: 10.1016/0025-5564(84)90059-2.

[12]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510. doi: 10.4310/CMS.2009.v7.n2.a12.

[13]

T. R. Malthus, An Essay on the Principle of Population, St. Paul's London, 1798.

[14]

A. G. Mckendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130.

[15]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[16]

P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697-702. doi: 10.1016/j.crma.2004.03.006.

[17]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001.

[18]

R. A. Neumïler and J. A. Knoblich, Dividing cellular asymmetry: Asymmetric cell division and its implications for stem cells and cancer, Genes Dev., 23 (2009), 2675-2699.

[19]

B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, Journal of Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018.

[20]

T. Suebcharoen, B. Van-Brunt and G. C. Wake, Asymmetric cell division in a size-structured growth model, Differential and Integral Equations, 24 (2011), 787-799.

[21]

B. Van-Brunt, G. C. Wake and H. K. Kim, A singular Sturm-Liouville problem involving an advanced functional differential equation, European Journal of Applied Mathematics, 12 (2001), 625-644. doi: 10.1017/S0956792501004624.

[22]

B. Van-Brunt and M. Vlieg-Hulstman, An eigenvalue problem involving a functional differential equation arising in a cell growth model, ANZIAM J., 51 (2010), 383-393. doi: 10.1017/S1446181110000866.

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