American Institute of Mathematical Sciences

Advanced
2014, 11(2): 363-384. doi: 10.3934/mbe.2014.11.363

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, Japan
2. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received  November 2012 Revised  August 2013 Published  October 2013

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: renewal equations, age structured population models, Transient amplifying cell population dynamics, generation progression model, stochastic simulation..
Mathematics Subject Classification: 92B05, 92C37, 92D25, 37N2.
Citation: Shinji Nakaoka, Hisashi Inaba. Demographic modeling of transient amplifying cell population growth. Mathematical Biosciences & Engineering, 2014, 11 (2) : 363-384. doi: 10.3934/mbe.2014.11.363
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 (): 17.  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 (): 721.  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.

show all references

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 (): 17.  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 (): 721.  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.

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