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.  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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