A model for asymmetrical cell division

Previous Article
Optimality and stability of symmetric evolutionary games with applications in genetic selection
MBE Home
This Issue
Next Article
The effect of time delay in plant--pathogen interactions with host demography

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

Abstract
Related Papers

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), 329-357. 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.

[1]	Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[2]	Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[3]	Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[4]	Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793
[5]	Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338
[6]	Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[7]	Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291
[8]	Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[9]	Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[10]	Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure and Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
[11]	Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[12]	Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033
[13]	Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[14]	John A. D. Appleby, Denis D. Patterson. Subexponential growth rates in functional differential equations. Conference Publications, 2015, 2015 (special) : 56-65. doi: 10.3934/proc.2015.0056
[15]	Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 309-323. doi: 10.3934/jcd.2021013
[16]	Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[17]	Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[18]	Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[19]	Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[20]	Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

2018 Impact Factor: 1.313

Tools

Metrics

Other articles
by authors

on AIMS
Ali Ashher Zaidi Bruce Van Brunt Graeme Charles Wake
on Google Scholar
Ali Ashher Zaidi Bruce Van Brunt Graeme Charles Wake

[Back to Top]