The impact of maturation time distributions on the structure and growth of cellular populations

Asma Alshehri; John Ford; Rachel Leander; Asma Alshehri; John Ford; Rachel Leander

doi:10.3934/mbe.2020098

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1855-1888. doi: 10.3934/mbe.2020098

Previous Article Next Article

Research article Special Issues

The impact of maturation time distributions on the structure and growth of cellular populations

1.
3404 Bakr Bin Mobashar Street, Taiba Box 8036, Jeddah 23833, KSA
2.
Department of Computer Science, Middle Tennessee State University, MTSU Box 48, Murfreesboro, TN 37132, USA
3.
Department of Mathematical Sciences, Middle Tennessee State University, MTSU Box 34, Murfreesboro, TN 37132, USA

Received: 30 September 2019 Accepted: 01 December 2019 Published: 18 December 2019

Here we study how the structure and growth of a cellular population vary with the distribution of maturation times from each stage. We consider two cell cycle stages. The first represents early G1. The second includes late G1, S, G2, and mitosis. Passage between the two reflects passage of an important cell cycle checkpoint known as the restriction point. We model the population as a system of partial differential equations. After establishing the existence of solutions, we characterize the maturation rates and derive the steady-state age and stage distributions as well as the asymptotic growth rates for models with exponential and inverse Gaussian maturation time distributions. We find that the stable age and stage distributions, transient dynamics, and asymptotic growth rates are substantially different for these two maturation models. We conclude that researchers modeling cellular populations should take care when choosing a maturation time distribution, as the population growth rate and stage structure can be heavily impacted by this choice. Furthermore, differences in the models' transient dynamics constitute testable predictions that can help further our understanding of the fundamental process of cellular proliferation. We hope that our numerical methods and programs will provide a scaffold for future research on cellular proliferation.
Citation: Asma Alshehri, John Ford, Rachel Leander. The impact of maturation time distributions on the structure and growth of cellular populations[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1855-1888. doi: 10.3934/mbe.2020098

Related Papers:

Abstract

Here we study how the structure and growth of a cellular population vary with the distribution of maturation times from each stage. We consider two cell cycle stages. The first represents early G1. The second includes late G1, S, G2, and mitosis. Passage between the two reflects passage of an important cell cycle checkpoint known as the restriction point. We model the population as a system of partial differential equations. After establishing the existence of solutions, we characterize the maturation rates and derive the steady-state age and stage distributions as well as the asymptotic growth rates for models with exponential and inverse Gaussian maturation time distributions. We find that the stable age and stage distributions, transient dynamics, and asymptotic growth rates are substantially different for these two maturation models. We conclude that researchers modeling cellular populations should take care when choosing a maturation time distribution, as the population growth rate and stage structure can be heavily impacted by this choice. Furthermore, differences in the models' transient dynamics constitute testable predictions that can help further our understanding of the fundamental process of cellular proliferation. We hope that our numerical methods and programs will provide a scaffold for future research on cellular proliferation.

References

[1]	P. Gabriel, S. P. Garbett, V. Quaranta, D. R. Tyson, G. F. Webb, The contribution of age structure to cell population responses to targeted therapeutics, J. Theor. Biol., 311 (2012), 19-27.
[2]	S. S. Hamed, R. M. Straubinger, W. J. Jusko, Pharmacodynamic modeling of cell cycle and apoptotic effects of gemcitabine on pancreatic adenocarcinoma cells, Cancer Chemother Pharmacol, 72 (2013), 553-563.
[3]	X. Miao, G. Koch, S. Ait-Oudhia, R. M. Straubinger, W. J. Jusko, Pharmacodynamic modeling of cell cycle effects for gemcitabine and trabected in combinations in pancreatic cancer cells, Front. Pharmacol., 7 (2016), 421.
[4]	J. A. Alberts Bruce, Molecular Biology of the Cell: a Problems Approach, 4th edition, Garland Science, New York, NY, 2002.
[5]	E. S. Wenzel, A. T. Singh, Cell-cycle checkpoints and aneuploidy on the path to cancer, In Vivo, 32 (2018), 1-5.
[6]	A. Zetterberg, O. Larsson, K. G. Wiman, What is the restriction point?, Curr. Opin. Cell Biol., 7 (1995), 835-842.
[7]	C. Schwarz, A. Johnson, M. K. oivomägi, E. Zatulovskiy, C. J. Kravitz, A. Doncic, et al., A precise Cdk activity threshold determines passage through the restriction point, Mol. Cell, 69 (2018), 253-264.
[8]	A. Zilman, V. Ganusov, A. Perelson, Stochastic models of lymphocyte proliferation and death, PLoS One, 5 (2010), e12775.
[9]	A. V. Gett, P. D. Hodgkin, A cellular calculus for signal integration by T cells, Nature, 1 (2000), 239-244.
[10]	K. Léon, J. Faro, J. Caneiro, A general mathematical framework to model generation structure in a population of asynchronously dividing cells, J. Theor. Biol., 229 (2004), 455-476.
[11]	R. Callard, P. Hodgkin, Modeling T- and B-cell growth and differentiation, Immunol. Rev., 216 (2007), 119-129.
[12]	J. A. Smith, L. Martin, Do cells cycle?, Proc. Natl. Acad. Sci. U.S.A., 70 (1973), 1263-1267.
[13]	A. Golubev, Exponentially modified Gaussian (EMG) relevance to distributions related to cell proliferation and differentiation, J. Theor. Biol., 262 (2010), 257-266.
[14]	A. Golubev, Genes at work in random bouts, BioEssays, 34 (2012), 311-319.
[15]	A. Golubev, Applications and implications of the exponentially modified gamma distribution as a model for time variabilities related to cell proliferation and gene expression, J. Theor. Biol., 393 (2016), 203-217.
[16]	S. J. Cain, P. C. Chau, Transition probability cell cycle model part I-balanced growth, J. Theor. Biol., 185 (1997), 55-67.
[17]	S. Svetina, B. Žekš, Transition probability model of the cell cycle exhibiting the age-distribution for cells in the indeterministic state of the cell cycle, in Biomathematics and Cell Kinetics (eds. A. J. Valleron and P. D. M. MacDonald), Elsevier/North-Holland Biomedical Press, New York, 1978, 71-82.
[18]	S. Cooper, The continuum model: statistical implications, J. Theor. Biol., 94 (1982), 783-800.
[19]	S. Banerjee, K. Lo, M. K. Daddysman, A. Selewa, T. Kuntz, A. R. Dinner, et al., Biphasic growth dynamics during Caulobacter crescentus division, bioRxiv, (2017), 047589.
[20]	Z. W. Jones, R. Leander, V. Quaranta, L. A. Harris, D. R. Tyson, A drift-diffusion checkpoint model predicts a highly variable and growth-factor-sensitive portion of the cell cycle g1 phase, PLOS ONE, 13 (2018), 1-20.
[21]	J. Folks, R. S. Chikara, The inverse Gaussian distribution and its statistical application-a review, J. R. Statist. Soc. B, 40 (1978), 263-289.
[22]	O. Angulo, J. López-Marcos, Numerical integration of nonlinear size-structured population equations, Ecol. Model., 133 (2000), 3-14.
[23]	S. C. Tate, S. Cai, R. T. Ajamie, T. Burke, R. P. Beckmann, E. M. Chan, et al., Semi-mechanistic pharmacokinetic/pharmacodynamic modeling of the antitumor activity of LY2835219, a new cyclin-dependent kinase 4/6 inhibitor, in mice bearing human tumor xenografts, Clin. Cancer Res., 20 (2014), 3763-3774.
[24]	R. R. Goldberg, Methods of Real Analysis, John Wiley & Sons, Hoboken, NJ, 1976.
[25]	K. Ito, F. Kappel, G. Peichel, A fully discretized approximation scheme for size-structured population models, SIAM J. Numer. Anal., 28 (1991), 923-954.
[26]	C. C. Pugh, Undergraduate texts in mathematics, in Real Mathematical Analysis (eds. S. Axler, F. Gehring and K. Ribet), vol. 19, Springer International Publishing, New York, NY, 2003.
[27]	Wikipedia, Leibniz integral rule, 2019. Available from: https://en.wikipedia.org/wiki/Leibniz_integral_rule.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)