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

Demographic modeling of transient amplifying cell population growth

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.
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.  doi: 10.1007/s11538-010-9524-5.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1038/nrm2636.  Google Scholar

[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.  doi: 10.1007/s11538-006-9094-8.  Google Scholar

[5]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge Monographs on Applied and Computational Mathematics, (2004).  doi: 10.1017/CBO9780511543234.  Google Scholar

[6]

R. Callard and P. Hodgkin, Modeling T- and B-cell growth and differentiation,, Immunol. Rev., 216 (2007), 119.   Google Scholar

[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.   Google Scholar

[8]

D. Donjerković and D. W. Scott, Activation-induced cell death in B lymphocytes,, Cell. Res., 10 (2000), 179.  doi: 10.1038/sj.cr.7290047.  Google Scholar

[9]

G. Feeney, Population dynamics based on birth intervals and parity progression,, Population Studies, 37 (1983), 75.   Google Scholar

[10]

H. Von Foerster, Some remarks on changing populations in The Kinetics of Cellular Proliferation,, Grune and Stratton, (1959).   Google Scholar

[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.   Google Scholar

[12]

A. V. Gett and P. D. Hodgkin, A cellular calculus for signal integration by T cells,, Nat. Immunol., 1 (2000), 239.   Google Scholar

[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.  doi: 10.1016/0021-9991(76)90041-3.  Google Scholar

[14]

M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence,, Math. Biosci., 86 (1987), 67.  doi: 10.1016/0025-5564(87)90064-2.  Google Scholar

[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.  doi: 10.1073/pnas.0905629106.  Google Scholar

[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.  doi: 10.1073/pnas.0700026104.  Google Scholar

[17]

H. Inaba, Duration-Dependent Multistate Population Dynamics,, Working Paper Series 9, (1992).   Google Scholar

[18]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Studies, 1 (1988), 49.  doi: 10.1080/08898488809525260.  Google Scholar

[19]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., 65 (2012), 309.  doi: 10.1007/s00285-011-0463-z.  Google Scholar

[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.  doi: 10.1016/j.mbs.2008.08.005.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2004.04.011.  Google Scholar

[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.  doi: 10.1006/jtbi.2001.2505.  Google Scholar

[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.  doi: 10.1007/s00285-006-0046-6.  Google Scholar

[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.  Google Scholar

[26]

N. MacDonald, Biological Delay System: Linear Stability Theory,, Cambridge Studies in Mathematical Biology, (1989).   Google Scholar

[27]

A. G. McKendrick, Application of mathematics to medical problems,, Proc. Edinburgh. Math. Soc., 44 (1926), 98.   Google Scholar

[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.  doi: 10.1007/s11538-011-9668-y.  Google Scholar

[29]

S. J. Morrison and J. Kimble, Asymmetric and symmetric stem-cell divisions in development and cancer,, Nature, 441 (2006), 1068.  doi: 10.1038/nature04956.  Google Scholar

[30]

S. J. Morrison and A. C. Spradling, Stem cells and niches: Mechanisms that promote stem cell maintenance throughout life,, Cell, 132 (2008), 598.  doi: 10.1016/j.cell.2008.01.038.  Google Scholar

[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.  doi: 10.1016/j.cam.2004.09.019.  Google Scholar

[32]

K. M. Murphy, Janeway's Immunobiology,, 8th Edition, (2012).   Google Scholar

[33]

S. Nakaoka and K. Aihara, Stochastic simulation of structured skin cell population dynamics,, J. Math. Biol., 66 (2013), 807.  doi: 10.1007/s00285-012-0618-6.  Google Scholar

[34]

A. Philpott and P. R. Yew, The xenopus cell cycle: An overview,, Methods Mol. Biol., 296 (2005), 95.   Google Scholar

[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.   Google Scholar

[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), (1978), 389.   Google Scholar

[37]

O. Scherbaum and G. Rasch, Cell size distribution and single cell growth in Tetrahymena pyriformis GL,, Acta Pathol. Microbiol. Scand., 41 (1957), 161.   Google Scholar

[38]

J. A. Smith and L. Martin, Do cells cycle?, Proc. Natl. Acad. Sci. USA, 70 (1973), 1263.  doi: 10.1073/pnas.70.4.1263.  Google Scholar

[39]

K. Soetaert, T. Petzoldt and R. W. Setzer, Solving differential equations in R: Package desolve,, Journal of Statistical Software, 33 (2010), 1.   Google Scholar

[40]

H. Takahasi and M. Mori, Double exponential formulas for numerical integration,, Publ. Res. Inst. Math. Sci., 9 (): 721.  doi: 10.2977/prims/1195192451.  Google Scholar

[41]

H. R. Thieme, Mathematics in Population Biology,, Princeton Series in Theoretical and Computational Biology, (2003).   Google Scholar

[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.  doi: 10.1137/S0036139995289842.  Google Scholar

[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.  Google Scholar

[44]

A. Zilman, V. V. Ganusov and A. S. Perelson, Stochastic models of lymphocyte proliferation and death,, PLoS One, 5 (2010).  doi: 10.1371/journal.pone.0012775.  Google Scholar

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.  doi: 10.1007/s11538-010-9524-5.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1038/nrm2636.  Google Scholar

[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.  doi: 10.1007/s11538-006-9094-8.  Google Scholar

[5]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge Monographs on Applied and Computational Mathematics, (2004).  doi: 10.1017/CBO9780511543234.  Google Scholar

[6]

R. Callard and P. Hodgkin, Modeling T- and B-cell growth and differentiation,, Immunol. Rev., 216 (2007), 119.   Google Scholar

[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.   Google Scholar

[8]

D. Donjerković and D. W. Scott, Activation-induced cell death in B lymphocytes,, Cell. Res., 10 (2000), 179.  doi: 10.1038/sj.cr.7290047.  Google Scholar

[9]

G. Feeney, Population dynamics based on birth intervals and parity progression,, Population Studies, 37 (1983), 75.   Google Scholar

[10]

H. Von Foerster, Some remarks on changing populations in The Kinetics of Cellular Proliferation,, Grune and Stratton, (1959).   Google Scholar

[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.   Google Scholar

[12]

A. V. Gett and P. D. Hodgkin, A cellular calculus for signal integration by T cells,, Nat. Immunol., 1 (2000), 239.   Google Scholar

[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.  doi: 10.1016/0021-9991(76)90041-3.  Google Scholar

[14]

M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence,, Math. Biosci., 86 (1987), 67.  doi: 10.1016/0025-5564(87)90064-2.  Google Scholar

[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.  doi: 10.1073/pnas.0905629106.  Google Scholar

[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.  doi: 10.1073/pnas.0700026104.  Google Scholar

[17]

H. Inaba, Duration-Dependent Multistate Population Dynamics,, Working Paper Series 9, (1992).   Google Scholar

[18]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Studies, 1 (1988), 49.  doi: 10.1080/08898488809525260.  Google Scholar

[19]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., 65 (2012), 309.  doi: 10.1007/s00285-011-0463-z.  Google Scholar

[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.  doi: 10.1016/j.mbs.2008.08.005.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2004.04.011.  Google Scholar

[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.  doi: 10.1006/jtbi.2001.2505.  Google Scholar

[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.  doi: 10.1007/s00285-006-0046-6.  Google Scholar

[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.  Google Scholar

[26]

N. MacDonald, Biological Delay System: Linear Stability Theory,, Cambridge Studies in Mathematical Biology, (1989).   Google Scholar

[27]

A. G. McKendrick, Application of mathematics to medical problems,, Proc. Edinburgh. Math. Soc., 44 (1926), 98.   Google Scholar

[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.  doi: 10.1007/s11538-011-9668-y.  Google Scholar

[29]

S. J. Morrison and J. Kimble, Asymmetric and symmetric stem-cell divisions in development and cancer,, Nature, 441 (2006), 1068.  doi: 10.1038/nature04956.  Google Scholar

[30]

S. J. Morrison and A. C. Spradling, Stem cells and niches: Mechanisms that promote stem cell maintenance throughout life,, Cell, 132 (2008), 598.  doi: 10.1016/j.cell.2008.01.038.  Google Scholar

[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.  doi: 10.1016/j.cam.2004.09.019.  Google Scholar

[32]

K. M. Murphy, Janeway's Immunobiology,, 8th Edition, (2012).   Google Scholar

[33]

S. Nakaoka and K. Aihara, Stochastic simulation of structured skin cell population dynamics,, J. Math. Biol., 66 (2013), 807.  doi: 10.1007/s00285-012-0618-6.  Google Scholar

[34]

A. Philpott and P. R. Yew, The xenopus cell cycle: An overview,, Methods Mol. Biol., 296 (2005), 95.   Google Scholar

[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.   Google Scholar

[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), (1978), 389.   Google Scholar

[37]

O. Scherbaum and G. Rasch, Cell size distribution and single cell growth in Tetrahymena pyriformis GL,, Acta Pathol. Microbiol. Scand., 41 (1957), 161.   Google Scholar

[38]

J. A. Smith and L. Martin, Do cells cycle?, Proc. Natl. Acad. Sci. USA, 70 (1973), 1263.  doi: 10.1073/pnas.70.4.1263.  Google Scholar

[39]

K. Soetaert, T. Petzoldt and R. W. Setzer, Solving differential equations in R: Package desolve,, Journal of Statistical Software, 33 (2010), 1.   Google Scholar

[40]

H. Takahasi and M. Mori, Double exponential formulas for numerical integration,, Publ. Res. Inst. Math. Sci., 9 (): 721.  doi: 10.2977/prims/1195192451.  Google Scholar

[41]

H. R. Thieme, Mathematics in Population Biology,, Princeton Series in Theoretical and Computational Biology, (2003).   Google Scholar

[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.  doi: 10.1137/S0036139995289842.  Google Scholar

[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.  Google Scholar

[44]

A. Zilman, V. V. Ganusov and A. S. Perelson, Stochastic models of lymphocyte proliferation and death,, PLoS One, 5 (2010).  doi: 10.1371/journal.pone.0012775.  Google Scholar

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