A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process

  • Mathisca C. M. de Gunst
  • Willem R. van Zwet
Open Access
Part of the Selected Works in Probability and Statistics book series (SWPS)


De Gunst has formulated a stochastic model for the growth of a certain type of plant cell population that initially consists of n cells. The total cell number N n (t) as predicted by the model is a non-Markovian counting process. The relative growth of the population, n −1(N n (t) - n), converges almost surely uniformly to a nonrandom function X. In the present paper we investigate the behavior of the limit process X(t) as t tends to infinity and determine the order of magnitude of the duration of the process N n (t). There are two possible causes for the process N n to stop growing, and correspondingly, the limit process X(t) has a derivative X’(t) that is the product of two factors, one or both of which may tend to zero as t tends to infinity. It turns out that there is a remarkable discontinuity in the tail behavior of the processes. We find that if only one factor of X’(t) tends to zero, then the rate at which the limit process reaches its final limit is much faster and the order of magnitude of the duration of the process N n is much smaller than when both occur approximately at the same time.

Key words and phrases

Stochastic model population growth non-Markovian counting process tail behavior duration 


  1. ALBERTS, B., BRAY, D., LEWIS, J., RAFF, M., ROBERTS, K. and WATSON, J. D. (1989). Molecular Biology of the Cell, 2nd ed. Garland, New York.Google Scholar
  2. BARBOUR, A. D. (1975). The duration of a closed stochastic epidemic. Biometrika 62 477-482.CrossRefMATHMathSciNetGoogle Scholar
  3. BAYLISS, M. W. (1985). The regulation of the cell division cycle in cultured plant cells. In The Cell Division Cycle in Plants (J. A. Bryant and D. Francis, eds.). Cambridge Univ. Press.Google Scholar
  4. BROOKS, R. F., BENNETT, D. C. and SMITH, J. A. (1980). Mammalian cells need two random transitions. Cell 19 493- 504.CrossRefGoogle Scholar
  5. CASTOR, F. A. L. (1980). A G1 rate model accounts for cell-cycle kinetics attributed to 'transition probability.' Nature 287 857- 859.CrossRefGoogle Scholar
  6. COOPER, S. (1982). The continuum model: statistical implications. J. Theoret. Biol. 94 783-800.CrossRefGoogle Scholar
  7. DE GUNST, M. C. M. (1989). A Random Model for Plant Cell Population Growth. CWJ Tract 58. Math. Centrum, Amsterdam.Google Scholar
  8. DE GUNST, M. C. M., HARKES, P. A. A., VAL, J., VAN ZWET, W. R. and LIBBENGA, K. R. (1990). Modelling the growth of a batch culture of plant cells: a corpuscular approach. Enzyme Microb. Techno!. 12 61-71.Google Scholar
  9. DE GUNST, M. C. M. and VAN ZWET, W. R. (1992). A non-Markovian model for cell population growth: speed of convergence and central limit theorem. Stochastic Process. Appl. 41 297-324.Google Scholar
  10. HoEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30.CrossRefMATHMathSciNetGoogle Scholar
  11. KURTZ, T. G. (1982). Representation and approximation of counting processes. In Advances in Filtering and Optimal Stochastic Control Lecture Notes in Control and Inform. Sci. 42 177-191. Springer, New York.Google Scholar
  12. MORRIS, P., SCRAGG, A. H., STAFFORD, A. and FOWLER, M. W., EDS. (1986). Secondary Metabolism in Plant Cell Cultures. Cambridge Univ. Press.Google Scholar
  13. NAGAEV, A. V. and MUKHOMOR, T. P. (1975). A limit distribution of the duration of an epidemic. Theory Probab. Appl. 20 805- 818.Google Scholar
  14. NELSON, S. and GREEN, P. J. (1981). The random transition mode l of the cell cycle. A critical review. Cancer Chemother. Pharmacal. 6 11-18.Google Scholar
  15. ROELS, J. A. (1983). Energetics and Kinetics in Biotechnology. North-Holland, Amsterdam.Google Scholar
  16. SHIELDS, R. (1977). Transition probability and the origin of variation in the cell cycle. Nature 267 704-707.CrossRefGoogle Scholar
  17. SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.Google Scholar
  18. SMITH, J. A. and MARTIN, L. (1973). Do cells cycle? Proc. Nat. Acad. Sci. U.S.A. 70 1263- 1267.Google Scholar
  19. STREET, H. E. (1973). Plant Tissue and Cell Culture. Botanical Monographs 11. Blackwell, Oxford.Google Scholar
  20. TREWAVAS, A. J. (1985). Growth substances, calcium and the regulation of cell division. In The Cell Division Cycle in Plants. (J. A. Bryant and D. Francis, eds.). Cambridge Univ. Press.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Mathisca C. M. de Gunst
    • 1
    • 2
    • 3
    • 4
    • 5
    • 6
  • Willem R. van Zwet
    • 1
    • 2
    • 3
    • 4
    • 5
    • 6
  1. 1.Free University of AmsterdamAmsterdamThe Netherlands
  2. 2.University of LeidenLeidenThe Netherlands
  3. 3.University of North CarolinaNorth CarolinaUSA
  4. 4.Department of MathematicsFree UniversityAmsterdamThe Netherlands
  5. 5.Department of MathematicsUniversity of LeidenLeidenThe Netherlands
  6. 6.Department of StatisticsUniversity of North CarolinaChapel HillUSA

Personalised recommendations