# American Institute of Mathematical Sciences

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. 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
 [1] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & 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 & Continuous Dynamical Systems - A, 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 & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014 [4] Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793 [5] Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291 [6] Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993 [7] 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 & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115 [8] Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338 [9] Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016 [10] Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033 [11] Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27 [12] 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 [13] Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097 [14] Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481 [15] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [16] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [17] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [18] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [19] Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345 [20] Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283

2018 Impact Factor: 1.313