Bulletin of Mathematical Biology

, Volume 74, Issue 2, pp 300–326 | Cite as

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Original Article

Abstract

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.

Keywords

CFSE-labeling Cell cycle Age-dependent multitype branching process Cyton model Smith–Martin model Differential equation model Agent-based model Hybrid optimization Parameter estimation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akaike, H. (1973). Information theory as an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), Second international symposium on information theory (pp. 267–281). Budapest: Akademiai Kiado. Google Scholar
  2. Akkouchi, M. (2005). On the convolution of gamma distributions. Soochow J. Math., 31(2), 205–211. MathSciNetMATHGoogle Scholar
  3. Asquith, B., Debacq, C., Florins, A., Gillet, N., Sanchez-Alcaraz, T., Mosley, A., & Willems, L. (2006). Quantifying lymphocyte kinetics in vivo using carboxyfluorescein diacetate succinimidyl ester (CFSE). Proc. R. Soc. B, 273, 1165–1171. CrossRefGoogle Scholar
  4. Athreya, K. B., & Ney, P. E. (1972). Branching processes. Berlin: Springer. MATHGoogle Scholar
  5. Bellman, R., & Harris, T. (1952). On age-dependent binary branching processes. Ann. Math., 55(2), 280–295. MathSciNetMATHCrossRefGoogle Scholar
  6. Bernard, S., Pujo-Menjouret, L., & Mackey, M. C. (2003). Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys. J., 84, 3414–3424. CrossRefGoogle Scholar
  7. Bonnevier, J. L., & Mueller, D. L. (2002). Cutting edge: B7/CD28 interactions regulate cell cycle progression independent of the strength of TCR signaling. J. Immunol., 169(12), 6659–6663. Google Scholar
  8. Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociol. Methods Res., 33, 261–304. MathSciNetCrossRefGoogle Scholar
  9. Clyde, R. G., Bown, J. L., Hupp, T. R., Zhelev, N., & Crawford, J. W. (2006). The role of modelling in identifying drug targets for diseases of the cell cycle. J. R. Soc. Interface, 22, 617–627. CrossRefGoogle Scholar
  10. Cooper, S. (1982). The continuum model: statistical implications. J. Theor. Biol., 94, 783–800. CrossRefGoogle Scholar
  11. Cowan, R., & Morris, V. B. (1986). Cell population dynamics during the differentiation phase of tissue development. J. Theor. Biol., 122, 205–224. MathSciNetCrossRefGoogle Scholar
  12. Crump, K. S., & Mode, C. J. (1969). An age-dependent branching process with correlations among sister cells. J. Appl. Probab., 6, 205–219. MathSciNetMATHCrossRefGoogle Scholar
  13. De Boer, R. J., & Perelson, A. S. (2005). Estimating division and death rates from CFSE data. J. Comput. Appl. Math., 184, 140–164. MathSciNetMATHCrossRefGoogle Scholar
  14. De Boer, R. J., Homann, D., & Perelson, A. S. (2003). Different dynamics of CD4+ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection. J. Immunol., 171(8), 3928–3935. Google Scholar
  15. De Boer, R. J., Ganusov, V. V., Milutinovic, D., Hodgkin, P. D., & Perelson, A. S. (2006). Estimating lymphocyte division and death rates from CFSE data. Bull. Math. Biol., 68, 1011–1031. CrossRefGoogle Scholar
  16. Deenick, E. K., Hasbold, J., & Hodgkin, P. D. (1999). Switching to IgG3, IgG2b, and IgA is division linked and independent, revealing a stochastic framework for describing differentiation. J. Immunol., 163, 4707–4714. Google Scholar
  17. Deenick, E. K., Gett, A. V., & Hodgkin, P. D. (2003). Stochastic model of T cell proliferation: a calculus revealing IL-2 regulation of precursor frequencies, cell cycle time, and survival. J. Immunol., 170(10), 4963–4972. Google Scholar
  18. Fleurant, C., Duchesne, J., & Raimbault, P. (2004). An allometric model for trees. J. Theor. Biol., 227(1), 137–147. MathSciNetCrossRefGoogle Scholar
  19. Foulds, K. E., Zenewicz, L. A., Shedlock, D. J., Jiang, J., Troy, A. E., & Shen, H. (2002). Cutting edge: CD4 and CD8 T cells are intrinsically different in their proliferative responses. J. Immunol., 168, 1528–1532. Google Scholar
  20. Ganusov, V. V., Pilyugin, S. S., De Boer, R. J., Murali-Krishna, K., Ahmed, R., & Antia, R. (2005). Quantifying cell turnover using CFSE data. J. Immunol. Methods, 298, 183–200. CrossRefGoogle Scholar
  21. Ganusov, V. V., Milutinovic, D., & De Boer, R. J. (2007). IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: insights from modeling CFSE data. J. Immunol., 179, 950–957. Google Scholar
  22. Gett, A. V., & Hodgkin, P. D. (2000). A cellular calculus for signal integration by T cells. Nat. Immunol., 1(3), 239–244. CrossRefGoogle Scholar
  23. Glover, F. (1977). Heuristics for integer programming using surrogate constraints. Decis. Sci., 8, 156–166. CrossRefGoogle Scholar
  24. Guo, Z., & Tay, J. C. (2008). Multi-timescale event-scheduling in multi-agent immune simulation models. Biosystems, 91, 126–145. CrossRefGoogle Scholar
  25. Hasbold, J. A., Lyons, A. B., Kehry, M. R., & Hodgkin, P. D. (1998). Cell division number regulates IgG1 and IgE switching of B cells following stimulation by CD40 ligand and IL-4. Eur. J. Immunol., 28, 1040–1051. CrossRefGoogle Scholar
  26. Hawkins, E. D., Turner, M. L., Dowling, M. R., van Gend, C., & Hodgkin, P. D. (2007). A model of immune regulation as a consequence of randomized lymphocyte division and death times. Proc. Natl. Acad. Sci. USA, 104(12), 5032–5037. CrossRefGoogle Scholar
  27. Hawkins, E. D., Markham, J. F., McGuinness, L. P., & Hodgkin, P. D. (2009). A single-cell pedigree analysis of alternative stochastic lymphocyte fates. Proc. Natl. Acad. Sci. USA, 106(32), 13457–13462. CrossRefGoogle Scholar
  28. Heyde, C. C., & Seneta, E. (1977). I.J. Bienayme: statistical theory anticipated. Berlin: Springer. MATHGoogle Scholar
  29. Hodgkin, P. D., Lee, J. H., & Lyons, A. B. (1996). B cell differentiation and isotype switching is related to division cycle number. J. Exp. Med., 184, 277–281. CrossRefGoogle Scholar
  30. Hyrien, O., & Zand, M. S. (2008). A mixture model with dependent observations for the analysis of CFSE-labeling experiments. J. Am. Stat. Assoc., 103(481), 222–239. MathSciNetMATHCrossRefGoogle Scholar
  31. Hyrien, O., Mayer-Pröschel, M., Noble, M., & Yakovlev, A. (2005). A stochastic model to analyze clonal data on multi-type cell populations. Biometrics, 61, 199–207. MathSciNetMATHCrossRefGoogle Scholar
  32. Jagers, P. (1975). Branching processes with biological applications. London: Wiley. MATHGoogle Scholar
  33. Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes (2nd ed.). San Diego: Academic Press. MATHGoogle Scholar
  34. Kimmel, M. (1980). Cellular population dynamics. I. Model construction and reformulation. Math. Biosci., 48(3–4), 211–224. MathSciNetMATHCrossRefGoogle Scholar
  35. Kimmel, M., & Axelrod, D. E. (1991). Unequal cell division, growth regulation and colony size of mammalian cells: a mathematical model and analysis of experimental data. J. Theor. Biol., 153, 157–180. CrossRefGoogle Scholar
  36. Kimmel, M., & Axelrod, D. E. (2002). Branching processes in biology. New York: Springer. MATHGoogle Scholar
  37. Kimmel, M., & Traganos, F. (1986). Estimation and prediction of cell cycle specific effects of anticancer drugs. Math. Biosci., 80, 187–208. MATHCrossRefGoogle Scholar
  38. Koch, A. L. (1999). The re-incarnation, re-interpretation and re-demise of the transition probability model. J. Biotech., 71, 143–156. CrossRefGoogle Scholar
  39. Laguna, M., & Marti, R. (2003). Scatter search: methodology and implementations. Boston: Kluwer Academic. CrossRefGoogle Scholar
  40. Laguna, M., & Marti, R. (2005). Experimental testing of advanced scatter search designs for global optimization of multimodal functions. J. Glob. Optim., 33, 235–355. MathSciNetMATHCrossRefGoogle Scholar
  41. Lee, H. Y., & Perelson, A. S. (2008). Modeling T cell proliferation and death in vitro based on labeling data: generalizations of the Smith–Martin cell cycle model. Bull. Math. Biol., 70(1), 21–44. MathSciNetMATHCrossRefGoogle Scholar
  42. Lee, H., Hawkins, E., Zand, M. S., Mosmann, T., Wu, H., Hodgkin, P. D., & Perelson, A. S. (2009). Interpreting CFSE obtained division histories of B cells in vitro with Smith–Martin and cyton type models. Bull. Math. Biol., 71(7), 1649–1670. MathSciNetMATHCrossRefGoogle Scholar
  43. Leon, K., Faro, J., & Carneiro, J. (2004). A general mathematical framework to model generation structure in a population of asynchronously dividing cells. J. Theor. Biol., 229, 455–476. MathSciNetCrossRefGoogle Scholar
  44. Liang, H., Miao, H., & Wu, H. (2010). Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model. Ann. Appl. Stat., accepted. Google Scholar
  45. Liu, D., Yu, J., Chen, H., Reichman, R., Wu, H., & Jin, X. (2006). Statistical determination of threshold for cellular division in the CFSE-labeling assay. J. Immunol. Methods, 312(1–2), 126–136. CrossRefGoogle Scholar
  46. Lyons, A. B. (2000). Analyzing cell division in vivo and in vitro using flow cytometric measurement of CFSE dye dilution. J. Immunol. Methods, 243, 147–154. CrossRefGoogle Scholar
  47. Macken, C. A., & Perelson, A. S. (1988). Lecture notes in biomathematics: Vol. 76. Stem cell proliferation and differentiation: a multitype branching process model. New York: Springer. MATHGoogle Scholar
  48. Mathai, A. (1982). Storage capacity of a dam with gamma type inputs. Ann. Inst. Stat. Math., 34(1), 591–597. MathSciNetMATHCrossRefGoogle Scholar
  49. Miao, H., Dykes, C., Demeter, L. M., Cavenaugh, J., Park, S. Y., Perelson, A. S., & Wu, H. (2008). Modeling and estimation of kinetic parameters and replicative fitness of HIV-1 from flow-cytometry- based growth competition experiments. Bull. Math. Biol., 70(6), 1749–1771. MathSciNetMATHCrossRefGoogle Scholar
  50. Miao, H., Dykes, C., Demeter, L. M., & Wu, H. (2009). Differential equation modeling of HIV viral fitness experiments: model identification, model selection, and multi-model inference. Biometrics, 65(1), 292–300. MathSciNetMATHCrossRefGoogle Scholar
  51. Miao, H., Xia, X., Perelson, A. S., & Wu, H. (2010). Identifiability of nonlinear ODE models with applications in viral dynamics. SIAM Rev. (in press). Google Scholar
  52. Moles, C. G., Banga, J. R., & Keller, K. (2004). Solving nonconvex climate control problems: pitfalls and algorithm performances. Appl. Soft Comput., 5(1), 35–44. CrossRefGoogle Scholar
  53. Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Ann. Inst. Stat. Math., 37(3), 541–544. MathSciNetMATHCrossRefGoogle Scholar
  54. Nocedal, J., & Wright, S. J. (1999). Numerical optimization. New York: Springer. MATHCrossRefGoogle Scholar
  55. Nordon, R. E., Nakamura, M., Ramirez, C., & Odell, R. (1999). Analysis of growth kinetics by division tracking. Immunol. Cell Biol., 77, 523–529. CrossRefGoogle Scholar
  56. Novak, B., & Tyson, J. J. (1995). Quantitative analysis of a molecular model of mitotic control in fission yeast. J. Theor. Biol., 173, 283–305. CrossRefGoogle Scholar
  57. Novak, B., & Tyson, J. J. (1997). Modeling the control of DNA replication in fission yeast. Proc. Natl. Acad. Sci. USA, 94, 9147–9152. CrossRefGoogle Scholar
  58. Novak, B., & Tyson, J. J. (2004). A model for restriction point control of the mammalian cell cycle. J. Theor. Biol., 230, 563–579. MathSciNetCrossRefGoogle Scholar
  59. Pilyugin, S. S., Ganusov, V. V., Murali-Krishna, K., Ahmed, R., & Antia, R. (2003). The rescaling method for quantifying the turnover of cell population. J. Theor. Biol., 225, 275–283. MathSciNetCrossRefGoogle Scholar
  60. Powell, E. O. (1955). Some features of the generation times of individual bacteria. Biometrika, 42(1–2), 16–44. Google Scholar
  61. Revy, P., Sospedra, M., Barbour, B., & Trautmann, A. (2001). Functional antigen-independent synapses formed between T cells and dendritic cells. Nat. Immunol., 2(10), 925–931. CrossRefGoogle Scholar
  62. Rodriguez-Fernandez, M., Egea, J. A., & Banga, J. R. (2006). Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems. BMC Bioinform., 7, 483. CrossRefGoogle Scholar
  63. Schwarz, G. (1978). Estimating the dimensions of a model. Ann. Stat., 6, 461–464. MATHCrossRefGoogle Scholar
  64. Sim, C. H. (1991). Point processes with correlated gamma interarrival times. Stat. Probab. Lett., 15(2), 135–141. MathSciNetCrossRefGoogle Scholar
  65. Smith, J. A., & Martin, L. (1973). Do cells cycle? Proc. Natl. Acad. Sci. USA, 70, 1263–1267. CrossRefGoogle Scholar
  66. Smith, J. A., Laurence, D. J. R., & Rudland, P. S. (1981). Limitations of cell kinetics in distinguishing cell cycle models. Nature, 293, 648–650. CrossRefGoogle Scholar
  67. Stewart, T., Strijbosch, L. W. G., Moors, J. J. A., & Van Batenburg, P. (2007). A simple approximation to the convolution of gamma distributions. Tilburg University, Center for Economic Research. Google Scholar
  68. Storn, R., & Price, K. (1997). Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim., 11, 341–359. MathSciNetMATHCrossRefGoogle Scholar
  69. Thom, H. C. S. (1968). Approximate convolution of the gamma and mixed gamma distributions. Mon. Weather Rev., 96(12), 883–886. CrossRefGoogle Scholar
  70. Toni, T., Welch, D., Strelkowa, N., Ipsen, A., & Stumpf, M. P. H. (2009). Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface, 6(31), 187–202. CrossRefGoogle Scholar
  71. Tyrcha, J. (2001). Age-dependent cell cycle models. J. Theor. Biol., 213(1), 89–101. MathSciNetCrossRefGoogle Scholar
  72. Tyson, J. J. (1991). Modeling the cell division cycle: cdc2 and cycling interactions. Proc. Natl. Acad. Sci. USA, 88, 7328–7332. CrossRefGoogle Scholar
  73. Vellaisamy, P., & Upadhye, N. S. (2009). On the sums of compound negative binomial and gamma random variables. J. Appl. Probab., 46, 272–283. MathSciNetMATHCrossRefGoogle Scholar
  74. Wellard, C., Markham, J., Hawkins, E. D., & Hodgkin, P. D. (2010). The effect of correlations on the population dynamics of lymphocytes. J. Theor. Biol., 264(2), 443–449. CrossRefGoogle Scholar
  75. Whitmire, J. K., & Ahmed, R. (2000). Costimulation in antiviral immunity: differential requirements for CD4(+) and CD8(+) T cell responses. Curr. Opin. Immunol., 12(4), 448–455. CrossRefGoogle Scholar
  76. Yakovlev, A. Y., & Yanev, N. M. (1989). Transient processes in cell proliferation kinetics. Heidelberg: Springer. MATHGoogle Scholar
  77. Yakovlev, A. Y., & Yanev, N. M. (2006). Branching stochastic processes with immigration in analysis of renewing cell pupulations. Math. Biosci., 203, 37–63. MathSciNetMATHCrossRefGoogle Scholar
  78. Yakovlev, A. Y., Mayer-Pröschel, M., & Noble, M. (1998). A stochastic model of brain cell differentiation in tissue culture. J. Math. Biol., 37, 49–60. MATHCrossRefGoogle Scholar
  79. Yakovlev, A. Y., Stoimenova, V. K., & Yanev, N. M. (2008). Branching processes as models of progenitor cell populations and estimation of the offspring distributions. J. Am. Stat. Assoc., 103(484), 1357–1366. MathSciNetCrossRefGoogle Scholar
  80. Ye, Y. (1987). Interior algorithms for linear, quadratic and linearly constrained non-linear programming. Ph.D. thesis, Dept. of ESS, Stanford University. Google Scholar
  81. Zilman, A., Ganusov, V. V., & Perelson, A. S. (2010). Stochastic models of lymphocyte proliferation and death. PLoS ONE, 5(9), e12775. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  • Hongyu Miao
    • 1
  • Xia Jin
    • 2
    • 3
  • Alan S. Perelson
    • 4
  • Hulin Wu
    • 1
  1. 1.Department of Biostatistics and Computational BiologyUniversity of Rochester School of Medicine and DentistryRochesterUSA
  2. 2.Department of MedicineUniversity of Rochester School of Medicine and DentistryRochesterUSA
  3. 3.Department of Microbiology and ImmunologyUniversity of Rochester School of Medicine and DentistryRochesterUSA
  4. 4.Theoretical Biology and Biophysics Group, MS-K710Los Alamos National LaboratoryLos AlamosUSA

Personalised recommendations