American Institute of Mathematical Sciences

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

A model for asymmetrical cell division

Ali Ashher Zaidi 1, , Bruce Van Brunt 2, and  Graeme Charles Wake 1,

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.
Keywords: eigenfunctions, Functional differential equations, hyperbolic partial differential equations., cell biology.
Mathematics Subject Classification: Primary: 34K08, 35L02, 34L10; Secondary: 92C3.
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.  doi: 10.1007/s00285-003-0254-2.  Google Scholar

[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.   Google Scholar

[3]

R. Begg, Cell-population Growth Modeling and Functional Differential Equations,, Ph.D thesis, (2007).   Google Scholar

[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.  doi: 10.1093/imammb/dqi015.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar

[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.  doi: 10.1007/PL00001379.  Google Scholar

[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.  doi: 10.1007/BF00277748.  Google Scholar

[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.  doi: 10.1017/S0334270000006366.  Google Scholar

[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.  doi: 10.1007/BF00160330.  Google Scholar

[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.  doi: 10.1016/0025-5564(84)90059-2.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, Lecture Notes in Biomathematics, (1986).  doi: 10.1007/978-3-662-13159-6.  Google Scholar

[16]

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

[17]

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

[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.   Google Scholar

[19]

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

[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.   Google Scholar

[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.  doi: 10.1017/S0956792501004624.  Google Scholar

[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.  doi: 10.1017/S1446181110000866.  Google Scholar

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.  doi: 10.1007/s00285-003-0254-2.  Google Scholar

[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.   Google Scholar

[3]

R. Begg, Cell-population Growth Modeling and Functional Differential Equations,, Ph.D thesis, (2007).   Google Scholar

[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.  doi: 10.1093/imammb/dqi015.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar

[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.  doi: 10.1007/PL00001379.  Google Scholar

[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.  doi: 10.1007/BF00277748.  Google Scholar

[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.  doi: 10.1017/S0334270000006366.  Google Scholar

[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.  doi: 10.1007/BF00160330.  Google Scholar

[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.  doi: 10.1016/0025-5564(84)90059-2.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, Lecture Notes in Biomathematics, (1986).  doi: 10.1007/978-3-662-13159-6.  Google Scholar

[16]

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

[17]

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

[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.   Google Scholar

[19]

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

[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.   Google Scholar

[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.  doi: 10.1017/S0956792501004624.  Google Scholar

[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.  doi: 10.1017/S1446181110000866.  Google Scholar

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