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ASYMMETRICAL CELL DIVISION WITH EXPONENTIAL GROWTH

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  04 June 2021

A. A. ZAIDI Open the ORCID record for A. A. ZAIDI [Opens in a new window]  and
B. VAN BRUNT Open the ORCID record for B. VAN BRUNT [Opens in a new window]
A. A. ZAIDI*
Affiliation:
Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan
B. VAN BRUNT
Affiliation:
School of Fundamental Sciences, Massey University, Palmerston North, New Zealand; [email protected].
*
Email: [email protected]
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Abstract

An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to “exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal.

Keywords

nonlocal partial differential equationsfunctional partial differential equations

MSC classification

Primary: 35R10: Partial functional-differential equations
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 1 , January 2021 , pp. 70 - 83
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Copyright
© Australian Mathematical Society 2021

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References

Ambartsumyan, V. A., “On the fluctuation of the brightness of the Milky Way”, Dokl. Akad. Nauk SSSR 44 (1944) 223–226, available at https://scholar.google.com/scholar_lookup?title=On+the+fluctuation+of+the+brightness+of+the+milky+way&author=Ambartsumian,+V.A.&publication_year=1994&journal=Dokl.+Akad.+Nauk+USSR&volume=44&pages=223-226.Google Scholar
Basse, B., Baguley, B., Marshell, E., Joseph, W., van Brunt, B., Wake, G. C. and Wall, D., “Modelling cell death in human tumor cell lines exposed to the anticancer drug paclitaxel”, J. Math. Biol. 49 (2004) 329–357; doi:10.1017/S0334270000006779.CrossRefGoogle ScholarPubMed
Begg, R., Wall, D. J. N. and Wake, G. C., “On a functional equation model of transient cell growth”, Math. Med. Biol. 22 (2005) 371–390; doi: 10.1093/imammb/dqi015.CrossRefGoogle ScholarPubMed
Derfel, G., van Brunt, B. and Wake, G. C., “A cell growth model revisited”, Funct. Differ. Equ. 19 (2012) 71–81, available at https://mro.massey.ac.nz/bitstream/handle/10179/9757/FDE_final.pdf?sequence=1&isAllowed=y.Google Scholar
Gaver, D. P., “An absorption probability problem”, J. Math. Anal. Appl. 9 (1964) 384–393; doi: 10.1016/0022-247X(64)90024-1.CrossRefGoogle Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation arising in modelling of cell growth”, J. Aust. Math. Soc. Ser. B (Currently, ANZIAM J.) 30 (1989) 424–435; doi:10.1017/S0334270000006366.CrossRefGoogle Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation determining steady size distributions for populations of cells growing exponentially”, J. Aust. Math. Soc. Ser. B (Currently, ANZIAM J.) 31 (1990) 344–353; doi:10.1017/S0334270000006779.Google Scholar
Hall, A. J., Wake, G. C. and Gandar, P. W., “Steady size distributions for cells in one dimensional plant tissues”, J. Math. Biol. 30 (1991) 101–123; doi:10.1007/BF00160330.CrossRefGoogle Scholar
Heijmans, H. J. A. M., “On the stable size distribution of populations reproducing by fission into two unequal parts”, Math. Biosci. 72 (1984) 19–50; doi:10.1016/0025-5564(84)90059-2.CrossRefGoogle Scholar
Iserles, A., “On the generalized pantograph functional differential equation”, European J. Appl. Math. 4 (1993) 1–38; doi:10.1017/S0956792500000966.CrossRefGoogle Scholar
Kato, T. and McLeod, J. B., “The functional differential equation ”, Bull. Amer. Math. Soc. 77 (1971) 891–937; doi:10.1090/S0002-9904-1971-12805-7.Google Scholar
Koch, A. L. and Schaechter, M., “A model for statistics of the cell division process”, J. Gen. Microbiol. 29 (1962) 435–454; doi: 10.1099/00221287-29-3-435.CrossRefGoogle Scholar
Metz, J. A. J. and Diekmann, O., The dynamics of physiologically structured populations, Volume 68 (Springer, Berlin, 1986) 1–511; doi: 10.1007/978-3-662-13159-6.CrossRefGoogle Scholar
Neumüller, R. A. and Knoblich, J. A., “Dividing cellular asymmetry: asymmetric cell division and its implications for stem cells and cancer”, Genes Dev. 23 (2009) 2675–2699; doi:10.1101/gad.1850809.CrossRefGoogle ScholarPubMed
Ockendon, J. and Tayler, A., “The dynamics of a current collection system for an electric locomotive”, Proc. R. Soc. A 322 (1971) 447–468; doi:10.1098/rspa.1971.0078.Google Scholar
Powell, E. O., “A note on Koch and Schaechter’s hypothesis about growth and fission of bacteria”, J. Gen. Microbiol. 37 (1964) 231–249; doi: 10.1099/00221287-37-2-231.CrossRefGoogle ScholarPubMed
Sinko, J. W. and Streifer, W., “A new model for age-size structure of a population”, Ecology 48 (1967) 910–918; doi:10.2307/1934533.CrossRefGoogle Scholar
Sinko, J. W. and Streifer, W., “A model for populations reproducing by fission”, Ecology 52 (1971) 330–335; doi:10.2307/1934592.CrossRefGoogle Scholar
Suebcharoen, T., van Brunt, B. and Wake, G. C., “Asymmetric cell division in a size-structured growth model’’, Differential Integral Equations 24 (2011) 787–799, available at https://projecteuclid.org/journals/differential-and-integral-equations/volume-24/issue-7_2f_8/Asymmetric-cell-division-in-a-size-structured-growth-model/die/1356628833.short. Google Scholar
Tyson, J. and Diekmann, O., “Sloppy size control of the division cycle”, J. Theoret. Biol. 118 (1986) 405–426; doi: 10.1016/S0022-5193(86)80162-X.CrossRefGoogle ScholarPubMed
van Brunt, B., Almalki, A., Lynch, T. and Zaidi, A., “On a cell division equation with a linear growth rate”, ANZIAM J. 59 (2018) 293–312; doi: 10.1017/S1446181117000591.Google Scholar
van Brunt and, B. Vlieg-Hulstman, M., “An eigenvalue problem involving a functional differential equation arising in a cell growth model”, ANZIAM J. 51 (2010) 383–393; doi: 10.1017/S1446181110000866.CrossRefGoogle Scholar
van Brunt, B. and Wake, G. C., “A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model”, European J. Appl. Math. 22 (2011) 151–168; doi: 10.1017/S0956792510000367.CrossRefGoogle Scholar
van Brunt, B., Wake, G. C. and Kim, H. K., “On a singular Sturm–Liouville problem involving an advanced functional differential equation”, European J. Appl. Math. 12 (2001) 625–644; doi: 10.1017/S0956792501004624.CrossRefGoogle Scholar
Wake, G. C., Cooper, S., Kim, H. K. and van Brunt, B., “Functional differential equations for cell-growth models with dispersion”, Commun. Appl. Anal. 4 (2000) 561–574, available at https://www.researchgate.net/publication/265494015_Functional_differential_equations_for_cell-growth_models_with_dispersion/link/593f5bec458515a6216b98ba/download.Google Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “Solutions to an advanced functional partial differential equation of the pantograph type”, Proc. R. Soc. A 471 (2015) 20140947; doi: 10.1098/rspa.2014.0947.CrossRefGoogle Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “Probability density function solutions to a Bessel type pantograph equation”, Appl. Anal. 95(11) (2015) 2565–2577; doi: 10.1080/00036811.2015.1102890.CrossRefGoogle Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “A model for asymmetrical cell division”, Math. Biosci. Eng. 12 (2015) 491–501; doi:10.3934/mbe.2015.12.491.CrossRefGoogle Scholar