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Journal of Mathematical Biology

, Volume 69, Issue 6–7, pp 1547–1583 | Cite as

Mathematical models for CFSE labelled lymphocyte dynamics: asymmetry and time-lag in division

  • Tatyana LuzyaninaEmail author
  • Jovana Cupovic
  • Burkhard Ludewig
  • Gennady Bocharov
Article

Abstract

Since their invention in 1994, fluorescent dyes such as carboxyfluorescein diacetate succinimidyl ester (CFSE) are used for cell proliferation analysis in flow cytometry. Importantly, the interpretation of such assays relies on the assumption that the label is divided equally between the daughter cells upon cell division. However, recent experimental studies indicate that division of cells is not perfectly symmetric and there is unequal distribution of protein between sister cell pairs. The uneven partition of protein or mass to daughter cells can lead to an overlap in the generations of CFSE-labelled cells with straightforward consequences for the resolution of individual generations. Numerous mathematical models developed so far for the analysis of CFSE proliferation assay incorporate the premise that the CFSE fluorescence intensity is halved in the two daughter cells. Here, we propose a novel modelling approach for the analysis of the CFSE cell proliferation assays which are characterized by poorly resolved peaks of cell generations in flow cytometric histograms. We formulate a mathematical model in the form of a system of delay hyperbolic partial differential equations which provides a good agreement with the CFSE histograms time-series data and allows an analytical treatment. The model is a further generalization of the recently proposed class of division- and label-structured models as it considers an asymmetric cell division. In addition, the basic structure of the cell cycle, i.e. the resting and cycling cell compartments, is taken into account. The model is used to estimate fundamental parameters such as activation rate, duration of the cell cycle, apoptosis rate, CFSE decay rate and asymmetry factor in cell division of monoclonal T cells during cognate interaction with dendritic cells.

Keywords

Division- and label-structured cell population dynamics  Delay hyperbolic PDE model Asymmetric cell division Inverse problem 

Mathematics Subject Classification (2000)

35R30 92-08 92C37 

Notes

Acknowledgments

The authors acknowledge the support of this work provided by the Swiss National Science Foundation, the Von-Tobel Foundation (Zurich), the Russian Foundation of Basic Research (11-01-00117a), the Programme of the Russian Academy of Sciences (Basic research for Medicine) and by the Swedish Institute, Visby Program.

References

  1. Akbarian V, Wang W, Audet J (2012) Measurement of generation-dependent proliferation rates and death rates during mouse erythroid progenitor cell differentiation. Cytometry A 81(5):382–389CrossRefGoogle Scholar
  2. Andrew SM, Baker CTH, Bocharov GA (2007) Rival approaches to mathematical modelling in immunology. J Comput Appl Math 205:669–686CrossRefzbMATHMathSciNetGoogle Scholar
  3. Baker CTH, Bocharov GA, Paul CAH, Rihan FA (2005) Computational modelling with functional differential equations: identification, selection and sensitivity. Appl Numer Math 53:107–129CrossRefzbMATHMathSciNetGoogle Scholar
  4. Banks HT, Thompson WC (2012) Mathematical models of dividing cell populations: application to CFSE data. Math Model Nat Phenom 7(5):24–52Google Scholar
  5. Banks HT, Sutton KL, Thompson WC, Bocharov G, Roose D, Schenkel T, Meyerhans A (2011a) Estimation of cell proliferation dynamics using CFSE data. Bull Math Biol 70:116–150Google Scholar
  6. Banks HT, Sutton KL, Thompson WC, Bocharov G, Doumic M, Schenkel T, Argilaguet J, Giest S, Peligero C, Meyerhans A (2011b) A new model for the estimation of cell proliferation dynamics using CFSE data. J Immunol Methods 373:143–160CrossRefGoogle Scholar
  7. Banks HT, Thompson WC, Peligero C, Giest S, Argilaguet J, Meyerhans A (2012) A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assay. CRSC-TR12-03, North Carolina State UniversityGoogle Scholar
  8. Banks HT, Kapraun DF, Thompson WC, Peligero C, Argilaguet J, Meyerhans A (2013a) A novel statistical analysis and interpretation of flow cytometry data. J Biol Dyn 7(1):96–132CrossRefGoogle Scholar
  9. Banks HT, Choi A, Huffman T, Nardini J, Poag L, Thompson WC (2013b) Quantifying CFSE label decay in flow cytometry data. Appl Math Lett 26(5):571–577CrossRefzbMATHMathSciNetGoogle Scholar
  10. Bergmann CC, Lane TE, Stohlman SA (2006) Coronavirus infection of the central nervous system: host-virus stand-off. Nat Rev Microbiol 4(2):121–132CrossRefGoogle Scholar
  11. Bernard S, Pujo-Menjouet L, Mackey MC (2003) Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys J 84(5):3414–3424CrossRefGoogle Scholar
  12. Burnham KP, Anderson DR (2002) Model selection and multimodel inference–a practical information-theoretic approach, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  13. Chang JT, Palanivel VR, Kinjyo I, Schambach F, Intlekofer AM, Banerjee A, Longworth SA, Vinup KE, Mrass P, Oliaro J, Killeen N, Orange JS, Russell SM, Weninger W, Reiner SL (2007) Asymmetric T lymphocyte division in the initiation of adaptive immune responses. Science 315(5819):1687–1691CrossRefGoogle Scholar
  14. Ciocca ML, Barnett BE, Burkhardt JK, Chang JT, Reiner SL (2012) Cutting edge: asymmetric memory T cell division in response to rechallenge. J Immunol 188(9):4145–4148CrossRefGoogle Scholar
  15. De Boer RJ, Perelson AS (2005) Estimating division and death rates from CFSE data. J Comput Appl Math 184:140–164CrossRefzbMATHMathSciNetGoogle Scholar
  16. De Boer RJ, Perelson AS (2013) Quantifying T lymphocyte turnover. J Theor Biol 327:45–87CrossRefGoogle Scholar
  17. De Boer RJ, Oprea M, Antia R, Murali-Krishna K, Ahmed R, Perelson AS (2001) Recruitment times, proliferation, and apoptosis rates during the \(\text{ CD8 }^+\) T-cell response to lymphocytic choriomeningitis virus. J Virol 75(22):10663–10669CrossRefGoogle Scholar
  18. Fernandes RL, Nierychlo M, Lundin L, Pedersen AE, Puentes Tellez PE, Dutta A, Carlquist M, Bolic A, Schpper D, Brunetti AC, Helmark S, Heins AL, Jensen AD, Nopens I, Rottwitt K, Szita N, van Elsas JD, Nielsen PH, Martinussen J, Srensen SJ, Lantz AE, Gernaey KV (2011) Experimental methods and modeling techniques for description of cell population heterogeneity. Biotechnol Adv 29(6):575–599CrossRefGoogle Scholar
  19. Ganusov VV, Pilyugin SS, de Boer RJ, Murali-Krishna K, Ahmed R, Antia R (2005) Quantifying cell turnover using CFSE data. J Immunol Methods 298(1–2):183–200CrossRefGoogle Scholar
  20. Gershenfeld N (2002) The nature of mathematical modelling. Cambridge University Press, CambridgeGoogle Scholar
  21. Gyllenberg M (1986) The size and scar distributions of the yeast Saccharomyces cerevisiae. J Math Biol 24:81–101CrossRefzbMATHMathSciNetGoogle Scholar
  22. Hadamard J (1932) Le probléme de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann, ParisGoogle Scholar
  23. Hasenauer J, Schittler D, Allgöwer F (2012a) A computational model for proliferation dynamics of division- and label-structured populations. arXiv:1202.4923v1[q-bio.PE]Google Scholar
  24. Hasenauer J, Schittler D, Allgöwer F (2012b) Analysis and simulation of division- and label-structured population models: a new tool to analyze proliferation assays. Bull Math Biol 74(11):2692–2732zbMATHMathSciNetGoogle Scholar
  25. Hawkins ED, Turner ML, Dowling MR, van Gend C, Hodgkin PD (2007) A model of immune regulation as a consequence of randomized lymphocyte division and death times. Proc Natl Acad Sci USA 104(12):5032–5037CrossRefGoogle Scholar
  26. Kendall DG (1948) On the role of variable generation time in the development of a stochastic birth process. Biometrika 35:316–330CrossRefzbMATHMathSciNetGoogle Scholar
  27. Knuth, KH (2006) Optimal data-based binning for histograms. arXiv:physics/0605197 [physics.data-an]Google Scholar
  28. Ko KH, Odell R, Nordon RE (2007) Analysis of cell differentiation by division tracking cytometry. Cytometry A 71(10):773–782CrossRefGoogle Scholar
  29. Kosarev EL, Pantos E (1983) Optimal smoothing of ‘noisy’ data by fast Fourier transform. J Phys E Sci Instrum 16:537–543CrossRefGoogle Scholar
  30. Lee HY, Hawkins E, Zand MS, Mosmann T, Wu H, Hodgkin PD, Perelson AS (2009) Interpreting CFSE obtained division histories of B cells in vitro with Smith–Martin and cyton type models. Bull Math Biol 71(7):1649–1670CrossRefzbMATHMathSciNetGoogle Scholar
  31. Ludewig B, Krebs P, Junt T, Metters H, Ford NJ, Anderson RM, Bocharov G (2004) Determining control parameters for dendritic cell-cytotoxic T lymphocyte interaction. Eur J Immunol 34:2407–2418CrossRefGoogle Scholar
  32. Luzyanina T, Mrusek S, Edwards JT, Roose D, Ehl S, Bocharov G (2007a) Computational analysis of CFSE proliferation assay. J Math Biol 54(1):57–89CrossRefzbMATHMathSciNetGoogle Scholar
  33. Luzyanina T, Roose D, Schenkel T, Sester M, Ehl S, Meyerhans A, Bocharov G (2007b) Numerical modelling of label-structured cell population growth using CFSE distribution data. Theor Biol Med Model 24:4–26Google Scholar
  34. Luzyanina T, Roose D, Bocharov G (2009) Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data. J Math Biol 59(5):581–603CrossRefzbMATHMathSciNetGoogle Scholar
  35. Lyons AB (2000) Analysing cell division in vivo and in vitro using flow cytometric measurement of CFSE dye dilution. J Immunol Methods 243(1–2):147–154CrossRefGoogle Scholar
  36. Lyons AB, Parish CR (1994) Determination of lymphocyte division by flow cytometry. J Immunol Methods 171(1):131–137CrossRefGoogle Scholar
  37. Mackey MC, Rudnicki R (1994) Global stability in a delayed partial differential equation describing cellular replication. J Math Biol 33:89–109CrossRefzbMATHMathSciNetGoogle Scholar
  38. Mantzaris NV (2006) Stochastic and deterministic simulations of heterogeneous cell population dynamics. J Theor Biol 241(3):690–706CrossRefMathSciNetGoogle Scholar
  39. Mantzaris NV (2007) From single-cell genetic architecture to cell population dynamics: quantitatively decomposing the effects of different population heterogeneity sources for a genetic network with positive feedback architecture. Biophys J 92(12):4271–4288CrossRefGoogle Scholar
  40. Mantzaris NV, Liou J, Daoutidis P, Srienc F (1999) Numerical solution of a mass structured cell population balance model in an environment of changing substrate concentration. J Biotechnol 71:157–174CrossRefGoogle Scholar
  41. Matera G, Lupi M, Ubezio P (2004) Heterogeneous cell response to topotecan in a CFSE-based proliferation test. Cytometry A 62(2):118–128CrossRefGoogle Scholar
  42. McKendrick AG (1925) Applications of mathematics to medical problems. Proc Edinb Math Soc 44:98–130CrossRefGoogle Scholar
  43. Metzger P (2012) A unified growth model for division-, age- and label-structured cell populations. University of Stuttgart, Stuttgart, Germany, Diploma ThesisGoogle Scholar
  44. Metzger P, Hasenauer J, Allgöwer F (2012) Modeling and analysis of division-, age-, and label-structured cell populations. In: Proceedings of the 9th workshop on computational systems biology (WCSB), vol 9, Ulm, GermanyGoogle Scholar
  45. Miao H, Jin X, Perelson AS, Wu H (2012) Evaluation of multitype mathematical models for CFSE-labeling experiment data. Bull Math Biol 74(2):300–326CrossRefzbMATHMathSciNetGoogle Scholar
  46. Monod J (1949) The growth of bacterial cultures. Ann Rev Microbiol 3:371–394CrossRefGoogle Scholar
  47. Nordon RE, Nakamura M, Ramirez C, Odell R (1999) Analysis of growth kinetics by division tracking. Immunol Cell Biol 77(6):523–529CrossRefGoogle Scholar
  48. Nordon RE, Ko KH, Odell R, Schroeder T (2011) Multi-type branching models to describe cell differentiation programs. J Theor Biol 277(1):7–18CrossRefGoogle Scholar
  49. Pagliara D, Savoldo B (2012) Cytotoxic T lymphocytes for the treatment of viral infections and posttransplant lymphoproliferative disorders in transplant recipients. Curr Opin Infect Dis 25(4):431–437CrossRefGoogle Scholar
  50. Pilyugin SS, Ganusov VV, Murali-Krishna K, Ahmed R, Antia R (2003) The rescaling method for quantifying the turnover of cell populations. J Theor Biol 225(2):275–83CrossRefMathSciNetGoogle Scholar
  51. Quah BJ, Parish CR (2012) New and improved methods for measuring lymphocyte proliferation in vitro and in vivo using CFSE-like fluorescent dyes. J Immunol Methods 379(1–2):1–14CrossRefGoogle Scholar
  52. Roederer M (2011) Interpretation of cellular proliferation data: avoid the panglossian. Cytometry A 79(2):95–101CrossRefGoogle Scholar
  53. Schittler D, Hasenauer J, Allgöwer F (2011) A generalized model for cellproliferation: Integrating division numbers and label dynamics. In: Proceedings of the eight international workshop on computationalsystems biology (WCSB, 2011), Zurich, Switzerland, pp 165–168Google Scholar
  54. Scott DW (1979) On optimal and data-based histograms. Biometrika 66(3):605–610CrossRefzbMATHMathSciNetGoogle Scholar
  55. Sennerstam R (1988) Partition of protein (mass) to sister cell pairs at mitosis: a re-evaluation. J Cell Sci 90(2):301–306Google Scholar
  56. Smith JA, Martin L (1973) Do cells cycle? Proc Natl Acad Sci USA 70(4):1263–1267CrossRefGoogle Scholar
  57. Sturges HA (1926) The choice of a class interval. J Am Stat Assoc 21(153):65–66CrossRefGoogle Scholar
  58. Taylor CC (1987) Akaike’s information criterion and the histogram. Biometrika 74(3):636–639CrossRefzbMATHMathSciNetGoogle Scholar
  59. Thompson WC (2011) Partial differential equation modelling of flow cytometry data from CFSE-based proliferation assays. PhD Dissertation. Department of Mathematics, North Carolina State University, RaleighGoogle Scholar
  60. Tikhonov AN, Arsenin VY (1977) Solutions of ill-posed problems. Winston & Sons, Washington, DCzbMATHGoogle Scholar
  61. Venzon DJ, Moolgavkar SH (1988) A method for computing profile-likelihood-based confidence intervals. Appl Stat 37(1):87–94CrossRefGoogle Scholar
  62. Wallace PK, Tario JD Jr, Fisher JL, Wallace SS, Ernstoff MS, Muirhead KA (2008) Tracking antigen-driven responses by flow cytometry: monitoring proliferation by dye dilution. Cytometry A 73(11):1019–1034CrossRefGoogle Scholar
  63. Wand MP (1997) Data-based choice of histograms bin width. Am Stat 51(1):59–64Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tatyana Luzyanina
    • 1
    Email author
  • Jovana Cupovic
    • 2
  • Burkhard Ludewig
    • 2
  • Gennady Bocharov
    • 3
  1. 1.Institute of Mathematical Problems in Biology, RASPushchinoRussia
  2. 2.Institute of ImmunobiologyKantonal Hospital St. GallenSt. GallenSwitzerland
  3. 3.Institute of Numerical Mathematics, RASMoscowRussia

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