Demographic modeling of transient amplifying cell population growth

Shinji Nakaoka; Hisashi Inaba

doi:10.3934/mbe.2014.11.363

Article Contents

2014, Volume 11, Issue 2: 363-384. Doi: 10.3934/mbe.2014.11.363 open access

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Demographic modeling of transient amplifying cell population growth

Shinji Nakaoka^1, and
Hisashi Inaba^2,

1.
Laboratory for Mathematical Modeling of Immune System, RCAI, RIKEN Center for Integrative Medical Sciences (IMS-RCAI), Suehiro-cho 1-7-22, Tsurumi-ku, Yokohama, 230-0045
2.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received: October 31, 2012

Revised: July 31, 2013

Published: September 2013

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Abstract

Quantitative measurement for the timings of cell division and death with the application of mathematical models is a standard way to estimate kinetic parameters of cellular proliferation. On the basis of label-based measurement data, several quantitative mathematical models describing short-term dynamics of transient cellular proliferation have been proposed and extensively studied. In the present paper, we show that existing mathematical models for cell population growth can be reformulated as a specific case of generation progression models, a variant of parity progression models developed in mathematical demography. Generation progression ratio (GPR) is defined for a generation progression model as an expected ratio of population increase or decrease via cell division. We also apply a stochastic simulation algorithm which is capable of representing the population growth dynamics of transient amplifying cells for various inter-event time distributions of cell division and death. Demographic modeling and the application of stochastic simulation algorithm presented here can be used as a unified platform to systematically investigate the short term dynamics of cell population growth.

Keywords:

Transient amplifying cell population dynamics,
generation progression model,
age structured population models,
renewal equations,
stochastic simulation.

Mathematics Subject Classification: 92B05, 92C37, 92D25, 37N25.

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References

[1]	H. T. Banks, K. L. Sutton, W. C. Thompson, G. Bocharov, D. Roose, T. Schenkel and A. Meyerhans, Estimation of cell proliferation dynamics using CFSE data, Bull. Math. Biol., 73 (2011), 116-150.doi: 10.1007/s11538-010-9524-5.
[2]	G. I. Bell and E. C. Anderson, Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351.
[3]	C. Blanpain and E. Fuchs, Epidermal homeostasis: A balancing act of stem cells in the skin, Nat. Rev. Mol. Cell. Biol., 10 (2009), 207-217.doi: 10.1038/nrm2636.
[4]	R. J. De Boer, V. V. Ganusov, D. Milutinović, P. D. Hodgkin and A. S. Perelson, Estimating lymphocyte division and death rates from CFSE data, Bull. Math. Biol., 68 (2006), 1011-1031.doi: 10.1007/s11538-006-9094-8.
[5]	H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monographs on Applied and Computational Mathematics, 15, Cambridge University Press, Cambridge, 2004.doi: 10.1017/CBO9780511543234.
[6]	R. Callard and P. Hodgkin, Modeling T- and B-cell growth and differentiation, Immunol. Rev., 216 (2007), 119-129.
[7]	E. K. Deenick, A. V. Gett and P. D. Hodgkin, Stochastic model of T cell proliferation: A calculus revealing Il-2 regulation of precursor frequencies, cell cycle time, and survival, J. Immunol., 170 (2003), 4963-4972.
[8]	D. Donjerković and D. W. Scott, Activation-induced cell death in B lymphocytes, Cell. Res., 10 (2000), 179-192.doi: 10.1038/sj.cr.7290047.
[9]	G. Feeney, Population dynamics based on birth intervals and parity progression, Population Studies, 37 (1983), 75-89.
[10]	H. Von Foerster, Some remarks on changing populations in The Kinetics of Cellular Proliferation, Grune and Stratton, 1959.
[11]	V. V. Ganusov, D. Milutinović and R. J. De Boer, Il-2 regulates expansion of CD4+ t cell populations by affecting cell death: Insights from modeling CFSE data, J. Immunol., 179 (2007), 950-957.
[12]	A. V. Gett and P. D. Hodgkin, A cellular calculus for signal integration by T cells, Nat. Immunol., 1 (2000), 239-244.
[13]	D. T. Gillespie, A general method for numerically simulation the stochastic time evolution of coupled chemical reactions, Journal of Computational Physics, 22 (1976), 403-434.doi: 10.1016/0021-9991(76)90041-3.
[14]	M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci., 86 (1987), 67-95.doi: 10.1016/0025-5564(87)90064-2.
[15]	E. D. Hawkins, J. F. Markham, L. P. McGuinness and P. D. Hodgkin, A single-cell pedigree analysis of alternative stochastic lymphocyte fates, Proc. Natl. Acad. Sci. USA, 106 (2009), 13457-13462.doi: 10.1073/pnas.0905629106.
[16]	E. D. Hawkins, M. L. Turner, M. R. Dowling, C. van Gend and P. D. Hodgkin, A model of immune regulation as a consequence of randomized lymphocyte division and death times, Proc. Natl. Acad. Sci. USA, 104 (2007), 5032-5037.doi: 10.1073/pnas.0700026104.
[17]	H. Inaba, Duration-Dependent Multistate Population Dynamics, Working Paper Series 9, Institute of Population Problems, Tokyo, 1992.
[18]	H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Studies, 1 (1988), 49-77.doi: 10.1080/08898488809525260.
[19]	H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.doi: 10.1007/s00285-011-0463-z.
[20]	H. Inaba and H. Nishiura, The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model, Math. Biosci., 216 (2008), 77-89.doi: 10.1016/j.mbs.2008.08.005.
[21]	J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, J. Math. Biol., 1 (1974/75), 17-36. doi: 10.1007/BF02339486.
[22]	K. León, J. Faro and J. Carneiro, A general mathematical framework to model generation structure in a population of asynchronously dividing cells, J. Theor. Biol., 229 (2004), 455-476.doi: 10.1016/j.jtbi.2004.04.011.
[23]	J. López-Sánchez, A. Murciano, R. Lahoz-Beltrá, J. Zamora, N. I. Giménez-Abián, J. F. López-Sáez, C. De La Torre and J. L. Cánovas, Modelling complex populations formed by proliferating, quiescent and quasi-quiescent cells: application to plant root meristems, J. Theor. Biol., 215 (2002), 201-213.doi: 10.1006/jtbi.2001.2505.
[24]	T. Luzyanina, S. Mrusek, J. T. Edwards, D. Roose, S. Ehl and G. Bocharov, Computational analysis of CFSE proliferation assay, J. Math. Biol., 54 (2007), 57-89.doi: 10.1007/s00285-006-0046-6.
[25]	T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans and G. Bocharov, Numerical modelling of label-structured cell population growth using CFSE distribution data, Theor. Biol. Med. Model., 4 (2007).doi: 10.1186/1742-4682-4-26.
[26]	N. MacDonald, Biological Delay System: Linear Stability Theory, Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1989.
[27]	A. G. McKendrick, Application of mathematics to medical problems, Proc. Edinburgh. Math. Soc., 44 (1926), 98-130.
[28]	H. Miao, X. Jin, A. S. Perelson and H. Wu, Evaluation of multitype mathematical models for CFSE-labeling experiment data, Bull. Math. Biol., 74 (2012), 300-326.doi: 10.1007/s11538-011-9668-y.
[29]	S. J. Morrison and J. Kimble, Asymmetric and symmetric stem-cell divisions in development and cancer, Nature, 441 (2006), 1068-1074.doi: 10.1038/nature04956.
[30]	S. J. Morrison and A. C. Spradling, Stem cells and niches: Mechanisms that promote stem cell maintenance throughout life, Cell, 132 (2008), 598-611.doi: 10.1016/j.cell.2008.01.038.
[31]	M. Muhammad, A. Nurmuhammad, M. Mori and M. Sugihara, Numerical solution of integral equations by means of the sinc-collocation method based on the double exponential transformation, J. Comput. Appl. Math., 177 (2005), 269-286.doi: 10.1016/j.cam.2004.09.019.
[32]	K. M. Murphy, Janeway's Immunobiology, 8th Edition, Immunobiology: The Immune System (Janeway), Garland Science, 2012.
[33]	S. Nakaoka and K. Aihara, Stochastic simulation of structured skin cell population dynamics, J. Math. Biol., 66 (2013), 807-835.doi: 10.1007/s00285-012-0618-6.
[34]	A. Philpott and P. R. Yew, The xenopus cell cycle: An overview, Methods Mol. Biol., 296 (2005), 95-112.
[35]	P. Revy, M. Sospedra, B. Barbour and A. Trautmann, Functional antigen-independent synapses formed between T cells and dendritic cells, Nat. Immunol., 2 (2001), 925-931.
[36]	S. I. Rubinow, Age-structured equations in the theory of cell populations, in Studies in Mathematical Biology Part II: Populations and Communities (ed. S. Levin), Studies in Mathematical Biology, 16, The Mathematical Association of America, Washington, D.C., 1978, 389-410.
[37]	O. Scherbaum and G. Rasch, Cell size distribution and single cell growth in Tetrahymena pyriformis GL, Acta Pathol. Microbiol. Scand., 41 (1957), 161-182.
[38]	J. A. Smith and L. Martin, Do cells cycle? Proc. Natl. Acad. Sci. USA, 70 (1973), 1263-1267.doi: 10.1073/pnas.70.4.1263.
[39]	K. Soetaert, T. Petzoldt and R. W. Setzer, Solving differential equations in R: Package desolve, Journal of Statistical Software, 33 (2010), 1-25.
[40]	H. Takahasi and M. Mori, Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci., 9 (1973/74), 721-741. doi: 10.2977/prims/1195192451.
[41]	H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.
[42]	G. S. K. Wolkowicz, H. Xia and S. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math., 57 (1997), 1281-1310.doi: 10.1137/S0036139995289842.
[43]	A. Yates, C. Chan, J. Strid, S. Moon, R. Callard, A. J. T. George and J. Stark, Reconstruction of cell population dynamics using CFSE, BMC Bioinformatics, 8 (2007).doi: 10.1186/1471-2105-8-196.
[44]	A. Zilman, V. V. Ganusov and A. S. Perelson, Stochastic models of lymphocyte proliferation and death, PLoS One, 5 (2010), e12775.doi: 10.1371/journal.pone.0012775.