Modelling cell generation times by using the tempered stable distribution

We show that the family of tempered stable distributions has considerable potential for modelling cell generation time data. Several real examples illustrate how these distributions can improve on currently assumed models, including the gamma and inverse Gaussian distributions which arise as special cases. Our applications concentrate on the generation times of oligodendrocyte progenitor cells and the yeast "Saccharomyces cerevisiae". Numerical inversion of the Laplace transform of the probability density function provides fast and accurate approximations to the tempered stable density, for which no closed form generally exists. We also show how the asymptotic population growth rate is easily calculated under a tempered stable model. Copyright (c) 2008 Royal Statistical Society.

[1]  M. Tuite,et al.  New Approximations to the Malthusian Parameter , 2006, Biometrics.

[2]  Tom Ross,et al.  Choosing probability distributions for modelling generation time variability , 1996 .

[3]  J Tyrcha.  Age-dependent cell cycle models. , 2001, Journal of theoretical biology.

[4]  Ananda Sen,et al.  The Theory of Dispersion Models , 1997, Technometrics.

[5]  M. Tuite,et al.  Estimating the number of prions in yeast cells. , 2004, Mathematical medicine and biology : a journal of the IMA.

[6]  Richard Cowan,et al.  A method for the measurement of variability in cell lifetimes , 1981 .

[7]  Ward Whitt,et al.  Probabilistic Scaling for the Numerical Inversion of Nonprobability Transforms , 1997, INFORMS J. Comput..

[8]  David G. Kendall,et al.  ON THE ROLE OF VARIABLE GENERATION TIME IN THE DEVELOPMENT OF A STOCHASTIC BIRTH PROCESS , 1948 .

[9]  Odd O. Aalen,et al.  Modelling Heterogeneity in Survival Analysis by the Compound Poisson Distribution , 1992 .

[10]  P. Green.  Modelling yeast cell growth using stochastic branching processes , 1981, Journal of Applied Probability.

[11]  J. Smith,et al.  Do cells cycle? , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[12]  T. A. Bromwich.  An Introduction To The Theory Of Infinite Series , 1908 .

[13]  Ward Whitt,et al.  An Introduction to Numerical Transform Inversion and Its Application to Probability Models , 2000 .

[14]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[15]  Gordon K. Smyth,et al.  Series evaluation of Tweedie exponential dispersion model densities , 2005, Stat. Comput..

[16]  J. Tyrcha.  Cell cycle progression. , 2004, Comptes rendus biologies.

[17]  Ollivier Hyrien,et al.  Stochastic modeling of oligodendrocyte generation in cell culture: model validation with time-lapse data , 2006, Theoretical Biology and Medical Modelling.

[18]  L. Hartwell,et al.  Unequal division in Saccharomyces cerevisiae and its implications for the control of cell division , 1977, The Journal of cell biology.

[19]  N. Shephard,et al.  Normal Modified Stable Processes , 2001 .

[20]  M. Samuels.  Distribution of the branching-process population among generations , 1971, Journal of Applied Probability.

[21]  K. Liang,et al.  Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions , 1987 .

[22]  Q. Vuong.  Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses , 1989 .

[23]  P. Iseger.  NUMERICAL TRANSFORM INVERSION USING GAUSSIAN QUADRATURE , 2005, Probability in the Engineering and Informational Sciences.

[24]  P. C. Chau,et al.  Transition probability cell cycle model. Part I--Balanced growth. , 1997, Journal of theoretical biology.

[25]  P. Hougaard.  Survival models for heterogeneous populations derived from stable distributions , 1986 .

[26]  M. Tuite,et al.  Approximations for Expected Generation Number , 2007, Biometrics.