Skip to main content

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 19))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abkowitz, J.L., Catlin, S.N. and Guttorp, P. 1996. Evidence that hematopoiesis may be a stochastic process in vivo. Nature Medicine 2: 190–197.

    Google Scholar 

  • Abramowitz, M. and Stegun, I.A. (eds.) 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th printing. U.S. Government Printing Office, Washington, D.C.

    MATH  Google Scholar 

  • Alexandersson, M. 1999. Branching processes and cell populations. Ph.D. thesis. Department of Mathematical Statistics, Chalmers University, Göteborg, Sweden.

    Google Scholar 

  • Alt, W. and Tyson, J.J. 1987. A stochastic model of cell division (with application to fission yeast). Mathematical Biosciences 83: 1–29.

    MathSciNet  Google Scholar 

  • Angerer W.P. 2001. An explicit representation of the Luria-Delbrück distribution. Journal of Mathematical Biology 42: 145–174.

    MATH  MathSciNet  Google Scholar 

  • Arino, O. and Kimmel, M. 1987. Asymptotic analysis of a cell cycle model based on unequal division. SIAM Journal of Applied Mathematics 47: 128–145.

    MATH  MathSciNet  Google Scholar 

  • Arino, O. and M. Kimmel. 1991. Asymptotic behavior of nonlinear semigroup describing a model of selective cell growth regulation. Journal of Mathematical Biology 29: 289–314.

    MATH  MathSciNet  Google Scholar 

  • Arino, O. and Kimmel, M. 1993. Comparison of approaches to modeling of cell population dynamics. SIAM Journal of Applied Mathematics 53: 1480–1504.

    MATH  MathSciNet  Google Scholar 

  • Arino O., Kimmel M. and Webb G.F. 1995. Mathematical modeling of the loss of telomere sequences. Journal of Theoretical Biology 177: 45–57.

    Google Scholar 

  • Arino, O., Kimmel, M. and Zerner, M. 1991. Analysis of a cell population model with unequal division and random transition. In: Mathematical Population Dynamics (Arino, O., Axelrod, D.E. and Kimmel, M. eds.). Marcel Dekker, New York, pp. 3–12.

    Google Scholar 

  • Arking, R. 1998. Biology of Aging. Sinauer, Sunderland, MA.

    Google Scholar 

  • Asmussen, S. and Hering, H. 1983. Branching Processes. Birkhauser, Boston, MA.

    MATH  Google Scholar 

  • Asteris, G. and Sarkar, S. 1996. Bayesian procedures for the estimation of mutation rates from fluctuation experiments. Genetics 142: 313–326.

    Google Scholar 

  • Athreya, K.B. and Ney, P.E. 1972. Branching Processes. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Axelrod, D.E. and Kuczek, T. 1989. Clonal heterogeneity in populations of normal and tumor cells. Computers and Mathematics with Applications 18: 871–881.

    Google Scholar 

  • Axelrod D.E., Baggerly, K.A. and Kimmel, M. 1994. Gene amplification by unequal sister chromatid exchange: Probabilistic modeling and analysis of drug resistance data. Journal of Theoretical Biology 168: 151–159.

    Google Scholar 

  • Axelrod, D.E., Gusev, Y. and Gamel, J.W. 1997. Ras-oncogene transformed and non-transformed cell population are each heterogeneous but respond differently to the chemotherapeutic drug cytosine arabinoside (Ara-C). Cancer Chemotherapy and Pharmacology 39: 445–451.

    Google Scholar 

  • Axelrod, D.E., Gusev, Y. and Kuczek, T. 1993. Persistence of cell cycle times over many generations as determined by heritability of colony sizes of ras oncogene-transformed and non-transformed cells. Cell Proliferation 26:235–249.

    Google Scholar 

  • Axelrod, D. E., Haider, F. R. and Tate, A. C. 1986. Distribution of interdivisional times in proliferating and differentiating Friend murine erythroleukaemia cells. Cell and Tissue Kinetics 19: 547–556.

    Google Scholar 

  • Baggerly, K.A. and Kimmel, M. 1995. Emergence of stable DNA repeats from random sequences under unequal sister chromatid exchange. In: Proceedings of the 1st World Congress of Nonlinear Analysts, Tampa, Florida, August 1992 (Lakshmikantham, V., ed.). Walter de Gruyter, Berlin, pp. 3409–3418.

    Google Scholar 

  • Bat, O., Kimmel, M. and Axelrod, D.E. 1997. Computer simulation of expansions of DNA triplet repeats in the fragile X syndrome and Huntington’s disease. Journal of Theoretical Biology 188: 53–67.

    Google Scholar 

  • Beaudry, A.A. and Joyce, G.F. 1992. Directed evolution of an RNA enzyme. Science 257: 635–641.

    Google Scholar 

  • Berndtsson, B. and Jagers, P. 1979. Exponential growth of a branching process usually implies stable age distribution. Journal of Applied Probability 16: 651–656.

    MATH  MathSciNet  Google Scholar 

  • Bertuzzi, A. and Gandolfi, A. 1983. Recent views on the cell cycle structure. Bulletin of Mathematical Biology 45: 605–616.

    MATH  Google Scholar 

  • Bertuzzi, A., Gandolfi, A., Giovenco, M. and Adelaide, M. 1981. Mathematical models of the cell cycle with a view to tumor studies. Mathematical Biosciences 53: 159–188.

    MATH  MathSciNet  Google Scholar 

  • Biggins, J.D. 1977. Chernoff’s theorem in the branching random walk. Journal of Applied Probability 14: 630–636.

    MATH  MathSciNet  Google Scholar 

  • Biggins, J.D. 1995. The growth and spread of the general branching random walk. Annals of Applied Probability 5: 1008–1024.

    MATH  MathSciNet  Google Scholar 

  • Biggins, J.D. 1997. How fast does a general branching random walk spread? In. Classical and Modern Branching Processes (Jagers, P. and Athreya, K., eds.) The IMA Volumes in Mathematics and Its Applications, 84. Springer-Verlag, Berlin, pp. 19–39.

    Google Scholar 

  • Biggins, J.D. and Kyprianou, A.E. 1996. Branching random walk: Seneta-Heyde norming. In: Trees (Chauvin, B., Cohen, S. and Rouault, A. eds.). Progress in Probability, 40. Birkhäuser, Basel, pp. 31–49.

    Google Scholar 

  • Biggins, J.D., Lubachevsky, B.D., Shwartz, A. and Weiss, A. 1991. A branching random walk with a barrier. The Annals of Applied Probability 1: 573–581.

    MATH  MathSciNet  Google Scholar 

  • Birky, C.W. and Skavaril, R.V. 1984. Random patitioning of cytoplasmic organelles at cell division: The effect of organelle and cell volume. Journal of Thoretical Biology 106: 441–447.

    Google Scholar 

  • Blackburn, E.H. 1991. Structure and function of telomeres. Nature 350: 569–573.

    Google Scholar 

  • Bobrowski, A. and Kimmel, M. 1999. Asymptotic behaviour of an operator exponential related to branching random walk models of DNA repeats. Journal of Biological Systems 7: 33–43.

    Google Scholar 

  • Borovkov, K.A. and Vatutin, V.A. 1977. Reduced critical branching processes in random environment. Stochastic Processes and Their Applications 77: 225–240.

    MathSciNet  Google Scholar 

  • Breiman, L. 1968. Probability. Addison-Wesley, Reading, MA.

    MATH  Google Scholar 

  • Brooks, R.F., Bennett, D.C. and Smith, J.A. 1980. Mammalian cell cycles need two random transitions. Cell 19: 493–504.

    Google Scholar 

  • Brown, P.C., Beverly, S.M. and R.T. Schimke, R.T. 1981. Relationship of amplified dihydrofolate reductase genes to double minute chromosomes in unstably resistant mouse fibroblasts cell lines. Molecular and Cellular Biology 1: 1077–1083.

    Google Scholar 

  • Caskey, C.T., Pizutti, A., Fu, Y.-H., Fenwick, R.G., Jr. and Nelson, D.L. 1992. Triplet repeat mutations in human disease. Science 256: 784–789.

    Google Scholar 

  • Chinnery P.F. and Turnbull, D.M. 1999. Mitochondrial DNA and disease. Lancet 354: 17–21.

    Google Scholar 

  • Ciampi, A., Kates, L., Buick, R., Kriukov, Y. and Till, J. E. 1986. Multi-type Galton-Watson process as a model for proliferating human tumour cell populations derived from stem cells: Estimation of stem cell self-renewal probabilities in human ovarian carcinomas. Cell Tissue Kinetics 19: 129–140.

    Google Scholar 

  • Cohn, H. and Klebaner, F. 1986. Geometric rate of growth in Markov Chains with applications to population-size-dependent models with dependent offspring. Stochastic Analysis and Applications 4: 283–308.

    MATH  MathSciNet  Google Scholar 

  • Coldman, A. J. 1987. Modeling resistance to cancer chemotherapeutic agents. In, Cancer Modeling (Thompson, J.R. and Brown, B.W., eds.). Marcel Dekker, Inc. New York, pp. 315–364.

    Google Scholar 

  • Coldman, A. J. and Goldie, A.J. 1983. A model for the resistance of tumor cells to cancer chemotherapeutic agents. Mathematical Biosciences 65: 291–307.

    MATH  Google Scholar 

  • Coldman, A. J. and Goldie, J. H. 1985. Role of mathematical modeling in protocol formulation in cancer chemotherapy. Cancer Treatment Reports 69: 1041–1048.

    Google Scholar 

  • Coldman, A. J. and Goldie, J. H. 1986. A stochastic model for the origin and treatment of tumors containing drug resistant cells. Bulletin of Mathematical Biology 48: 279–292.

    MATH  MathSciNet  Google Scholar 

  • Coldman, A. J., Goldie, J. H. and Ng, V. 1985. The effect of cellular differentiation on the development of permanent drug resistance. Mathematical Biosciences 74: 177–198.

    MATH  Google Scholar 

  • Cooper, S. 1979. A unifying model for the G1 period in prokaryotes and eukaryotes. Nature 280: 17–19.

    Google Scholar 

  • Cooper, S. 1984. The continuum model as a unified description of the division cycle of eukaryotes and prokaryotes. In: The Microbial Cell Cycle (Nurse, P. and Streiblova, E., eds.) CRC Press, Boca Raton, FL, pp. 8–27.

    Google Scholar 

  • Cooper, S. 1991. Bacterial Growth and Division: Biochemistry and Regulation of Prokaryotic and Eukaryotic Division Cycles. Academic Press, San Diego.

    Google Scholar 

  • Counter, C.M., Avilion, A.A., Lefeuvre, C.E., Stewart, N.G., Greider, C.W., Harley, C.B. and Bacchetti, S. 1992. Telomere shortening associated with chromosome instability is arrested in immortal cells which express telomerase activity. EMBO Journal 11: 1921–1929.

    Google Scholar 

  • Cowan R. 1985. Branching process results in terms of moments of the generationtime distribution. Biometrics 41: 681–689.

    MathSciNet  Google Scholar 

  • Cowan R. and Culpin D. 1981. A method for the measurement of variability in cell lifetimes. Mathematical Biosciences 54: 249–263.

    MATH  Google Scholar 

  • Cowan R. and Morris V.B. 1986. Cell population dynamics during the differentiation phase of tissue development. Journal of Theoretical Biology 122: 205–224.

    MathSciNet  Google Scholar 

  • Cowan, R. and Staudte, R. 1986. The bifurcating autoregression model in cell lineage studies. Biometrics 42: 769–783.

    MATH  Google Scholar 

  • Crump, K. S. 1970. On systems of renewal equations. Journal of Mathematical Analysis and Applications 30: 425–434.

    MATH  MathSciNet  Google Scholar 

  • Crump, K.S. and Hoel, D.G. 1974. Mathematical models for estimating mutation rates in cell populations. Biometrika 61: 237–252.

    MATH  MathSciNet  Google Scholar 

  • Crump, K. S. and Mode, C.J. 1969. An age-dependent branching process with correlations among sister cells. Journal of Applied Probability 6: 205–210.

    MATH  MathSciNet  Google Scholar 

  • Czerniak, B., Herz, F., Wersto, R.P. and Koss, L.G. 1992. Asymmetric distribution of oncogene products at mitosis. Proceedings of the National Academy of Sciences USA 89: 4860–4863.

    Google Scholar 

  • Darzynkiewicz, Z., Carter, S. and Kimmel, M. 1984. Effects of [3H]Udr on the cell-cycle progression of L1210 cells. Cell and Tissue Kinetics 17: 641–655.

    Google Scholar 

  • Darzynkiewicz, Z., Traganos, F. and Kimmel, M. 1986. Assay of cell cycle kinetics by multivariate flow cytometry using the principle of stathmokinesis. In: Techniques in Cell Cycle Analysis (Gray, J.E. and Darzynkiewicz, Z., eds.). Humana Press, Clifton, NJ, pp. 291–336.

    Google Scholar 

  • Darzynkiewicz, Z., Crissman, H., Traganos, F. and Steinkamp, J. 1982. Cell heterogeneity during the cell cycle. Journal of Cellular Physiology 113: 465–474.

    Google Scholar 

  • Darzynkiewicz, Z., Evenson, D. P., Staiano-Coico, L., Sharpless, T.K., Melamed, M. L. 1979. Correlation between cell cycle duration and RNA content. Journal of Cellular Physiology 100: 425–438.

    Google Scholar 

  • Dawson, D.A. and Hochberg, K.J. 1991. A multilevel branching model. Advances in Applied Probabiltiy 23: 701–715.

    MATH  MathSciNet  Google Scholar 

  • Dawson, D. and Perkins, E. 1991. Historical processes. Memoirs of the American Mathematical Society 93(454).

    Google Scholar 

  • Day, R. S. 1986a. Treatment sequencing, asymmetry, and uncertainty: Protocol strategies for combination chemotherapy. Cancer Research 46: 3876–3885.

    Google Scholar 

  • Day, R. 1986b. A branching process model for heterogeneous cell populations. Mathematical Biosciences 78: 73–90.

    MATH  MathSciNet  Google Scholar 

  • Demetrius, L., Schuster, P. and Sigmund, K. 1985. Polynuclotide evolution and branching processes. Bulletin of Mathematical Biology 47: 239–262.

    MATH  MathSciNet  Google Scholar 

  • Demos, J.P. 1982. Entertaining Satan: Witchcraft and the Culture of early New England. Oxford University Press, New York.

    Google Scholar 

  • Dibrov, B.F., Zhabotinsky, A.M., Neyfakh, Y.A., Orlova, M.P. and L.I. Churikova, L.I. 1985. Mathematical model of cancer chemotherapy. Periodic schedules of phase specific cytotoxic agent administration increasing the selectivity of therapy. Mathematical Biosciences 73: 1–31.

    MATH  MathSciNet  Google Scholar 

  • Dibrov, B.F., Zhabotinsky, A.M., Neyfakh, Y.A., Orlova, M.P. and Churikova, L.I. 1983. Optimal scheduling for cell synchronization by cycle-specific blockers. Mathematical Biosciences 66: 167–185.

    MATH  Google Scholar 

  • Doetsch, G. 1974. Introduction to the Theory and Application of the Laplace Transformation. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Durbin, R. Eddy, S. Krogh, A. and Mitchison, G. 1998. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids”. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Durrett, R. 1978. The genealogy of critical branching processes. Stochastic Processes and Their Applications 8: 101–116.

    MATH  MathSciNet  Google Scholar 

  • Etheridge, A.M. 1992. Conditioned superprocesses and a semilinear heat equation. In: Seminar on Stochastic Processes (Seattle, WA, 1992) (Cinlar, E., Chung, K.L. and Sharpe, M.J., eds.). Progress in Probability, 33. Birkhäuser, Boston, pp. 89–99.

    Google Scholar 

  • Falahati, A. 1999. Two-sex branching populations. Doctoral thesis. Department of Mathematical Statistics, Chalmers University, Göteborg, Sweden. Dissertations Series no. 1493.

    Google Scholar 

  • Fearn, D.H. 1972. Galton-Watson processes with generation dependence. Proceedings of the Sixth Berkley Symposium on Mathematical and Statistical Probabilitity 4: 159–172.

    MathSciNet  Google Scholar 

  • Fearn, D.H. 1976. Supercritical age dependent branching process with generation dependence. The Annals of Probability 4: 27–37.

    MATH  MathSciNet  Google Scholar 

  • Feller, W. 1968. An Introduction to Probability and Its Applications. Vol. 1, 3rd ed., Wiley, New York.

    MATH  Google Scholar 

  • Feller, W. 1971. An Introduction to Probability and Its Applications. Vol. 2, 2nd ed., Wiley, New York.

    MATH  Google Scholar 

  • Fleischmann, K. and Siegmund-Schultze, R. 1977. The structure of the reduced critical Galton-Watson processes. Mathematische Nachrichten 79: 233–241.

    MathSciNet  Google Scholar 

  • Fleischmann, K. and Vatutin, V.A. 1999. Reduced subcritical Galton-Watson processes in random environment. Advances in Applied Probability 31: 1–24.

    MathSciNet  Google Scholar 

  • Gawel, B. and Kimmel, M. 1996. Iterated Galton-Watson process. Journal of Applied Probability 33: 949–959.

    MATH  MathSciNet  Google Scholar 

  • Gillespie, J.H. 1986. Variability of evolutionary rates of DNA. Genetics 113: 1077–1091.

    Google Scholar 

  • Goldie, A.J. 1982. Drug resistance and chemotherapeutic strategy. In: Tumor Cell Heterogeneity (Owens, A.H., Coffey, D.S. and Baylin, S.B., eds.). Academic Press, New York, pp. 115–125.

    Google Scholar 

  • Goldie, J.H. and Coldman, A.J. 1979. A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treatment Reports 63: 1727–1733.

    Google Scholar 

  • Goldie, J.H. and Coldman, A.J. 1984. The genetic origin of drug resistance in neoplasms: Implications for systemic therapy. Cancer Research 44: 3643–3653.

    Google Scholar 

  • Goldie, J.H., Coldman, A.J. and Gudauskas, G.A. 1982. Rationale for the use of alternating non-cross-resistant chemotherapy. Cancer Treatment Reports 66: 439–449.

    Google Scholar 

  • González, M. and Molina, M. (1996) On the limit behaviour of a superadditive bisexual Galton-Watson branching process. Journal of Applied Probability 33: 960–967.

    MATH  MathSciNet  Google Scholar 

  • Greider, C.W. 1996. Telomere length regulation. Annual Review of Biochemistry 65: 337–365.

    Google Scholar 

  • Greider, C.W. and Blackburn, E.H. 1996. Telomeres, telomerase and cancer. Scientific American 274 (2): 92–97.

    Google Scholar 

  • Griffiths, R.C. and Tavaré, S. 1999. The ages of mutations in gene trees. Annals of Applied Probability 9: 567–590.

    MATH  MathSciNet  Google Scholar 

  • Gusev, Y. and Axelrod, D.E. 1995. Evaluation of models of inheritance of cell cycle times: Computer simulation and recloning experiments. In: Mathematical Population Dynamics: Analysis of Heterogeneity. Vol. 2 Carcinogenesis and Cell & Tumor Growth (Arino, A., Axelrod, D. and Kimmel, M., eds.). Wuerz Publishing, Winnipeg, Ontario, Canada, pp. 97–116.

    Google Scholar 

  • Guttorp P. 1991. Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics. Wiley, New York.

    Google Scholar 

  • Gyllenberg, M. 1986. The size and scar distributions of the yeast Saccharomyces cerevisiae. Journal of Mathematical Biology 24: 81–101.

    MATH  MathSciNet  Google Scholar 

  • Harley, C. B. 1991. Telomere loss: Mitotic clock or genetic time bomb? Mutation Research 256: 271–282.

    Google Scholar 

  • Harley, C.B. and Goldstein, S. 1980. Retesting the commitment theory of cellular aging. Science 207: 191–193.

    Google Scholar 

  • Harnevo, L.E. and Agur, Z. 1991. The dynamics of gene amplification described as a multitype compartmental model and as a branching process. Mathematical Biosciences 103: 115–138.

    MATH  MathSciNet  Google Scholar 

  • Harnevo, L.E. and Agur, Z. 1992. Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency. Cancer Chemotherapy and Pharmacology 30: 469–476.

    Google Scholar 

  • Harnevo, L.E. and Agur, Z. 1993. Use of mathematical models for understanding the dynamics of gene amplification. Mutation Research 292: 17–24.

    Google Scholar 

  • Harpending, H.C., Batzer, M.A., Gurven, M., Jorde, L.B., Rogers, A.R. and Sherry S.T. 1998. Genetic traces of ancient demography. Proceedings of the National Academy of Sciences USA 95: 1961–1967

    Google Scholar 

  • Harris, T.E. 1963. The Theory of Branching Processes. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Hasegawa, M. and Horai, S. 1990. Time of the deepest root for polymorphism in human mitochondrial DNA. Journal of Molecular Evolution 32: 37–42.

    Google Scholar 

  • Hästbacka, J., de la Chapelle, A., Kaitila, I., Sistonen, P., Weaver, A. and Lander, E. 1992. Linkage disequilibrium mapping in isolated founder populations: Diastrophic dysplasia in Finland. Nature Genetics 2: 204–211.

    Google Scholar 

  • Hejblum, G., Costagiola, D., Valleron, A.-J. and Mary, J.-Y. 1988. Cell cycle models and mother-daughter correlation. Journal of Theoretical Biology 131: 255–262.

    Google Scholar 

  • Hoel, D.G. and Crump K.S. 1974. Estimating the generation-time distribution of an age-dependent branching process. Biometrics 30: 125–135.

    MATH  MathSciNet  Google Scholar 

  • International Human Genome Sequencing Consortium. 2001. Initial sequencing and analysis of the human genome. Nature 409: 860–921.

    Google Scholar 

  • Jagers, P. 1975. Branching Processes with Biological Applications. Wiley, London.

    MATH  Google Scholar 

  • Jagers, P. 1983. Stochastic models for cell kinetics. Bulletin of Mathematical Biology 45: 507–519.

    MATH  MathSciNet  Google Scholar 

  • Jagers, P. 1991. The growth and stabilization of populations. Statistical Science 6: 269–283.

    MATH  MathSciNet  Google Scholar 

  • Jagers, P. 1992. Stabilities and instabilities in population dynamics. Journal Applied Probability 29: 770–780.

    MATH  MathSciNet  Google Scholar 

  • Jagers, P. 1995. Dependence in branching. Preprint 34: 1–17.

    Google Scholar 

  • Jagers, P. 2001. The deterministic evolution of general branching populations. IMS Lecture Notes and Monographs Series, 36: 384–398.

    MathSciNet  Google Scholar 

  • Jagers, P. and Nerman, O. 1996. The asymptotic composition of supercritical multitype branching populations (Mar Yor, ed.). Séeminaire de Probabilitées. Lecture Notes in Mathematics. Springer-Verlag, Berlin, pp. 40–54.

    Google Scholar 

  • Jagers, P. and Norrby, K. 1974. Estimation of the mean and variance of cycle times in cinemicrographically recorded cell populations during balanced exponential growth. Cell and Tissue Kinetics 7: 201–211.

    Google Scholar 

  • Joffe, A. and Waugh, W. 1982. Exact distributions of kin numbers in a Galton-Watson process. Journal Applied Probability 19: 767–775.

    MATH  MathSciNet  Google Scholar 

  • Joffe, A. and Waugh, W. 1985. The kin number problem in a multitype Galton-Watson population. Journal Applied Probabitlity 22: 37–47.

    MATH  MathSciNet  Google Scholar 

  • Joffe, A. and Waugh, W. 1986. Exact distributions of kin numbers in a multitype Galton-Watson process. In: Semi-Markov Models (Janssen, J., ed.). Plenum Press, New York, pp. 397–405.

    Google Scholar 

  • Jones, M.E. 1994. Luria-Delbrück fluctuation experiments; accounting simultaneously for plating efficiency and differential growth rate. Journal of Theoretical Biology 166: 355–363.

    Google Scholar 

  • Jones, M.E., Thomas, S.M. and Rogers, A. 1994. Luria-Delbrück fluctuation experiments: Design and analysis. Genetics 136: 1209–1216.

    Google Scholar 

  • Joyce, G.F. 1992. Directed molecular evolution. Scientific American 267(6) 90–97.

    Google Scholar 

  • Kaplan, N.L., Hill, W.G. and Weir, B.S. 1995. Likelihood methods for locating disease genes in nonequilibrium populations. American Journal of Human Genetics 56: 18–32.

    Google Scholar 

  • Kaufman, R.J., Brown, P.C. and Schimke, R.T. 1981. Loss and stabilization of amplified dihydrofolate reductase genes in mouse sarcoma S-180 cell lines. Molecular and Cellular Biology 1: 1084–1093.

    Google Scholar 

  • Kendal, W.S. and Frost, P. 1988. Pitfalls and practice of Luria-Delbrück fluctuation analysis: A review. Cancer Research 48: 1060–1065.

    Google Scholar 

  • Kesten, H. 1989. Supercritical branching processes with countably many types and the size of random Cantor sets. In: Probability, Statistics, and Mathematics: Papers in Honor of Samuel Karlin (Anderson, T.W., Athreya, K.B. and Iglehart, D.L., eds.). Academic Press, Boston, pp. 103–121.

    Google Scholar 

  • Kimmel, M. 1980a. Cellular population dynamics. I: Model construction and reformulation. Mathematical Biosciences 48: 211–224.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. 1980b. Cellular population dynamics. II: Investigation of solutions. Mathematical Biosciences 48: 225–239.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. 1980c. Time discrete model of cell population dynamics. Systems Science 6: 343–363.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. 1982. An equivalence result for integral equations with application to branching processes. Bulletin of Mathematical Biology 44: 1–15.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. 1983. The point process approach to age-and time-dependent branching processes. Advances in Applied Probability 15: 1–20.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. 1985 Nonparametric analysis of stathmokinesis. Mathematical Biosciences 74: 111–123.

    MATH  Google Scholar 

  • Kimmel, M. 1987. Metabolic events in the cell cycle of malignant and normal cells. A mathematical modeling approach. In: Cancer Modeling (Thompson, J.R. and Brown, B., eds.). Marcel Dekker, New York, pp. 215–235.

    Google Scholar 

  • Kimmel, M. 1994. Rapid genome evolution and cancer: A modeling perspective. Applied Mathematics and Computer Science 4: 163–177.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. 1997. Quasistationarity in a branching model of division-within-division. In: Classical and Modern Branching Processes (Athreya, K.B. and Jagers, P., eds.). IMA Volumes in Mathematics And Its Applications, 84. Springer-Verlag, New York, pp. 157–164.

    Google Scholar 

  • Kimmel, M. and Arino, O. 1991. Cell cycle kinetics with supramitotic control, two cell types and unequal division: A model of transformed embryonic cells. Mathematical Biosciences 105: 47–79.

    MATH  MathSciNet  Google Scholar 

  • Kimmel, M. and Axelrod, D.E. 1990. Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity. Genetics 125: 633–644.

    Google Scholar 

  • Kimmel, M. and Axelrod, D.E. 1991. Unequal cell division, growth regulation and colony size of mammalian cells: A mathematical model and analysis of experimental data. Journal of Theoretical Biology 153: 157–180.

    Google Scholar 

  • Kimmel, M. and Axelrod, D.E. 1994. Fluctuation test for two-stage mutations: Application to gene amplification. Mutation Research 306: 45–60.

    Google Scholar 

  • Kimmel, M. and Stivers, D. 1994. A time-continuous branching process model of unstable gene amplification. Bulletin of Mathematical Biology 56: 337–357.

    MATH  Google Scholar 

  • Kimmel, M. and Swierniak, A. 1982. On a certain optimal control problem related to the optimal chemotherapy of leukemia. Technical Reports of the Silesian Technical University, Series Automation (Zeszyty Naukowe Politechniki Slaskiej, Seria Automatyka) 65: 121–130. (in Polish, English abstract).

    Google Scholar 

  • Kimmel, M. and Traganos, F. 1985. Kinetic analysis of drug induced G2 block in vitro. Cell and Tissue Kinetics 18: 91–110.

    Google Scholar 

  • Kimmel, M. and Traganos, F. 1986. Estimation and prediction of cell cycle specific effects of anticancer drugs. Mathematical Biosciences 80: 187–208.

    MATH  Google Scholar 

  • Kimmel, M., Axelrod, D.E. and Wahl, G.M. 1992. A branching process model of gene amplification following chromosome breakage. Mutation Research 276: 225–239.

    Google Scholar 

  • Kimmel, M., Darzynkiewicz, Z. and Staiano-Coico, L. 1986. Stathmokinetic analysis of human epidermal cells in vitro. Cell and Tissue Kinetics 19: 289–304.

    Google Scholar 

  • Kimmel, M., Traganos, F. and Darzynkiewicz, Z. 1983. Do all daughter cells enter the ‘Indeterminate’ (‘A’) state of the cell cycle? Analysis of stathmokinetic experiment on L1210 cells. Cytometry 4: 191–201.

    Google Scholar 

  • Kimmel, M., Darzynkiewicz, Z., Arino, O. and Traganos, F. 1984. Analysis of a cell cycle model based on unequal division of metabolic constituents to daughter cells during cytokinesis. Journal of Theoretical Biology 110: 637–664.

    Google Scholar 

  • Kimmel, M., Grossi, A., Amuasi, J. and Vannucchi, A.M. 1990. Non-parametric analysis of platelet lifespan. Cell and Tissue Kinetics 23: 191–202.

    Google Scholar 

  • Kimmel M., Chakraborty, R., King, J.P., Bamshad, M., Watkins, W.S. and Jorde, L.B. 1998. Signatures of population expansion in microsatellite repeat data. Genetics 148: 1921–1930.

    Google Scholar 

  • Kirkwood, T.B.L. and Holliday, R. 1978. A stochastic model for the commitment of human cells to senescence. In: Biomathematics and Cell Kinetics (Valleron, A.J. and Macdonald, P.D., eds.). Elsevier/North-Holland, Amsterdam, pp. 161–172.

    Google Scholar 

  • Klebaner, F. 1988. Conditions for fixation of an allele in the density-dependent Wright-Fisher models. Journal of Applied Probability 25: 247–256.

    MATH  MathSciNet  Google Scholar 

  • Klebaner, F. 1990. Conditions for the unlimited growth in multitype population size dependent Galton-Watson processes. Bulletin of Mathematical Biology 52: 527–534.

    MATH  Google Scholar 

  • Klebaner, F. 1997. Population and density dependent branching process. In Classical and Modern Branching Processes (Athreya, K.B. and Jagers, P., eds.). IMA Volumes in Mathematics and Its Applications, 84. Springer, New York pp. 165–170.

    Google Scholar 

  • Klebaner, F. and Zeitouni, O. 1994. The exit problem for a class of density dependent branching systems. Annals of Applied Probability 4: 1188–1305.

    MATH  MathSciNet  Google Scholar 

  • Klein, B. and Macdonald, P.D.M. 1980. The multitype continuous-time Markov branching process in a periodic environment. Mathematical Sciences 12: 1–13.

    MathSciNet  Google Scholar 

  • Knolle, H. 1988. Cell Kinetic Modelling and the Chemotherapy of Cancer. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Koteeswaran, P. 1989. Estimating the age of a Galton-Watson process with binomial offspring distribution. Stochastic Analysis and Applications 7: 413–423.

    MATH  MathSciNet  Google Scholar 

  • Kotenko, J.L. Miller, J.H. and Robinson, A.I. 1987. The role of asymmetric cell division in Pteripodhyte cell differentiation. I. Localized metal accumulation and differentiation in Vittaria gemmae and Onoclea prothallia. Protoplasma 136: 81–95.

    Google Scholar 

  • Kowald, A. 1997. Possible mechanisms for the regulation of telomere length. Journal of Molecular Biology 273: 814–825.

    Google Scholar 

  • Kuczek, T. 1984. Stochastic modeling for the bacterial cell cycle. Mathematical Biosciences 69: 159–171.

    MATH  MathSciNet  Google Scholar 

  • Kuczek, T. and Axelrod, D.E. 1986. The importance of clonal heterogeneity and interexperiment variablity in modeling the eukaryotic cell cycle. Mathematical Biosciences 79: 87–96.

    Google Scholar 

  • Kuczek, T. and Axelrod, D.E. 1987. Tumor cell heterogeneity: Divided-colony assay for measuring drug response. Proceedings of the National Academy Sciences USA 84: 4490–4494.

    Google Scholar 

  • Kuczek, T. and Chan, T. C. K. 1988. Mathematical modeling for tumor resistance. Journal of the National Cancer Institute 80: 146–147. (Response: Goldie, J. H. and Coldman, A. J. 1988. Journal of the National Cancer Institute 80: 146–147.)

    Google Scholar 

  • Lapidus, R. 1984. Growth and division kinetics of asymmetrically dividing Tetrahymena thermophilia. Journal of Theoretical Biology 106: 135–140.

    Google Scholar 

  • Larson, D.D., Spangler, E.A. and Blackburn, E.H. 1987. Dynamics of telomere length variation in Tetrahymena thermophila. Cell 50: 477–483.

    Google Scholar 

  • Lea, D.E. and C.A. Coulson, C.A. 1949. The distribution of the numbers of mutants in bacterial populations. Journal of Genetics 49: 264–265.

    Google Scholar 

  • Levy, S.B. 1998. The challenge of antibiotic resistance. Scientific American 278(3): 46–53.

    Google Scholar 

  • Levy, M.Z., Allsopp, R.C., Futcher, A.B., Greider, C.W. and Harley, C.B. 1992. Teleomere end-replication problem and cell aging. Journal of Molecular Biology 225: 951–960.

    Google Scholar 

  • Lewontin, R.C. 2000. The Triple Helix: Gene, Organism, and Environment. Harvard University Press, Cambridge, MA.

    Google Scholar 

  • Lipow, C. 1975. A branching model with population size dependence. Advances in Applied Probability 7: 495–510.

    MATH  MathSciNet  Google Scholar 

  • Loeffler, M. and Wichmann, H.E. 1980. A comprehensive mathematical model of stem cell proliferation which reproduces most of the published experimental results. Cell and Tissue Kinetics 13: 543–561.

    Google Scholar 

  • Luria, S.E. and Delbrück, M. 1943. Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28: 491–511.

    Google Scholar 

  • Ma, W.T., Sandri, G. vH. and Sarkar, S. 1992. Analysis of the Luria-Delbrück distribution using discrete convolution powers. Journal of Applied Probability 29: 255–267.

    MATH  MathSciNet  Google Scholar 

  • Macdonald, P.D.M. 1978. Age distributions in the general cell kinetic model. In: Biomathematics and Cell Kinetics (Valleron, A.J. and Macdonald, P.D.M., eds.). Elsevier/North-Holland Biomedical Press, Amsterdam, pp. 3–20.

    Google Scholar 

  • Macken, C.A. and Perelson, A.S. 1985. Branching Processes Applied to Cell Surface Aggregaton Phenomena. Springer-Verlag, Berlin.

    Google Scholar 

  • Macken, C.A. and Perelson, A.S. 1988. Stem Cell Proliferation and Differentiation. A Multitype Branching Process Model. Lecture Notes in Biomathematics, 76. Springer-Verlag, Berlin.

    Google Scholar 

  • Mackillop, W.J. 1986. Instrinsic versus acquired drug resistance. Cancer Treatment Reports 70: 817. (Reply: Goldie, J.H. and Coldman, A.J. 1986. Cancer Treatment Reports 70: 818.)

    Google Scholar 

  • Maddox, J. 1992. Is molecular biology yet a science? Nature 355: 201.

    Google Scholar 

  • Metz, J.A.J. and Diekmann, O. (eds.). 1986. The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Mode, C.J. 1971. Multitype Branching Processes. Elsevier, New York.

    MATH  Google Scholar 

  • Morris, V.B. and Cowan, R. 1984. A growth curve of cell numbers in the neural retina of embryonic chicks. Cell and Tissue Kinetics 17: 199–208.

    Google Scholar 

  • Morris, V.B. and Taylor, I.W. 1985. Estimation of nonproliferating cells in the neural retina of embryonic chicks by flow cytometry. Cytometry 6: 375–380.

    Google Scholar 

  • Morrow, J. 1970. Genetic analysis of azaguanine resistance in an established mouse cell line. Genetics 65: 279–287.

    Google Scholar 

  • Moy, S.C. 1967. Extensions of a limit theorem of Everett, Ulam and Harris on multitype branching processes to a branching process with countably many types. Annals of Mathematical Statistics 38: 992–999.

    MATH  MathSciNet  Google Scholar 

  • Murnane, J.P. and Yezzi, M.J. 1988. Association of high rate of recombination with amplification of dominant selectable gene in human cells. Somatic Cell and Molecular Genetics 14: 273–286.

    Google Scholar 

  • Nagylaki, T. 1990. Models and approximations for random genetic drift. Theoretical Population Biology 37: 192–212.

    MATH  MathSciNet  Google Scholar 

  • Navidi, W., Tavare, S. and Arnheim, N. 1996. The role of the mutation rate and selective pressures on observed levels of the human mitochondrial DNA deletion mtDNA 4977. Unpublished manuscript.

    Google Scholar 

  • Nedelman, J., Downs, H. and Pharr, P. 1987. Inference for an age-dependent, multitype branching-process model of mast cells. Journal of Mathematical Biology 25: 203–226. (Erratum: Journal of Mathematical Biology 25: 571).

    MATH  MathSciNet  Google Scholar 

  • Neveu, J. 1975. Discrete-Parameter Martingales. rev. ed. Speed, T.P., North-Holland Mathematical Library, 10. North-Holland (Amsterdam/transl.). American Elsevier, New York.

    MATH  Google Scholar 

  • O’Connell, N. 1993. Yule process approximation for the skeleton of a branching process. Journal Applied Probability 30: 725–729.

    MATH  MathSciNet  Google Scholar 

  • O’Connell, N. 1995. The genealogy of branching processes and the age of our most recent common ancestor. Advances in Applied Probability 27: 418–442.

    MATH  MathSciNet  Google Scholar 

  • Olofsson, P. 1996. Branching processes with local dependencies. The Annals of Applied Probability 6: 238–268.

    MATH  MathSciNet  Google Scholar 

  • Olofsson, P. 2000. A branching process model of telomere shortening. Communications in Statistics. Stochastic Models 16: 167–177.

    MATH  MathSciNet  Google Scholar 

  • Olofsson, P. and Kimmel, M. 1999. Stochastic models of telomere shortening. Mathematical Biosciences 158: 75–92.

    MATH  MathSciNet  Google Scholar 

  • Olofsson, P. and Shaw, C. 2001. Exact sampling formulas for multi-type Galton-Watson processes. Journal of Mathematical Biology, to appear.

    Google Scholar 

  • Olofsson, P., Schwalb, O., Chakraborty, R., and Kimmel, M. 2001. An application of a general branching process in the study of the genetics of aging. Journal of Theoretical Biology 213: 547–557.

    Google Scholar 

  • Olovnikov, A.M. 1973. A theory of marginotomy. Journal of Theoretical Biology 41: 181–190.

    Google Scholar 

  • Pakes, A.G. 1993. Explosive Markov branching processes: Entrance laws and limiting behaviour. Advances in Applied Probability 25: 737–756.

    MATH  MathSciNet  Google Scholar 

  • Pakes, A.G. 1994. On the recognition & structure of probability generating functions. Research Report, Department of Mathematics, The University of Western Australia, Nedlands, WA, Australia. pp. 1–29.

    Google Scholar 

  • Pakes, A.G. 2000. Biological applications of branching processes. Research Report, Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA, Australia.

    Google Scholar 

  • Pakes, A.G. and Dekking, F.M. 1991. On family trees and subtrees of simple branching processes. Journal of Theoretical. Probability 4: 353–369.

    MATH  MathSciNet  Google Scholar 

  • Pakes, A.G. and Trajstman, A.C. 1985. Some properties of continuous-state branching processes, with applications to Bartoszynski’s virus model. Advances in Applied Probability 17: 23–41.

    MATH  MathSciNet  Google Scholar 

  • Pankratz, V.S. 1998. Stochastic Models and Linkage Disequilibrium. Doctoral thesis. Department of Statistics, Rice University, Houston, TX.

    Google Scholar 

  • Peterson, J.A. 1984. Analysis of variability in albumin content of sister hepatoma cells and model for geometric phenotypic variability (Quantitative Shift Model). Somatic Cell and Molecular Genetics 10: 345–357.

    Google Scholar 

  • Pharr, P.N., Nedelman, J., Downs, H.P., Ogawa, M. and Gross, A.J. 1985. A stochastic model for mast cell proliferation in culture. Journal of Cellular Physiology 125: 379–386.

    Google Scholar 

  • Polanski, A., Kimmel, M. and Swierniak A. 1997. Qualitative analysis of the infinite model of drug resistance evolution. In: Advances in Mathematical Population Dynamics — Molecules, Cells and Man (Arino, O., Axelrod, D. and Kimmel, M., eds.). World Scientific, Singapore, pp. 595–612.

    Google Scholar 

  • Polanski, A., Swierniak, A. and Duda, Z. 1993. Multiple solutions to the TPBVP arising in optimal scheduling of cancer chemotherapy. Conference Proceedings 1993, IEEE International Conference on Systems, Man and Cybernetics, Vol. 4, pp. 5–8.

    Google Scholar 

  • Puck, T.T. and Steffen, J. 1963. Life cycle analysis of mammalian cells. Part I. Biophysical Journal 3: 379–397.

    Google Scholar 

  • Richards, R.I. and Sutherland, G.R. 1994. Simple repeat DNA is not replicated simply. Nature Genetics 6: 114–116.

    Google Scholar 

  • Rigney, D.R. 1981. Correlation between the ages of sibling cell cycle events and a test of the “transition probabability” cell cycle model. In: Biomathematics and Cell Kinetics (Rotenberg, M., ed.). Elsevier/North Holland Biomedical Press, Amsterdam. pp. 157–166.

    Google Scholar 

  • Rittgen, W. 1983. Controlled branching processes and their applications to normal and malignant haematopoiesis. Bulletin of Mathematical Biology 45: 617–626.

    MATH  Google Scholar 

  • Rosen, R. 1986. Role of mathematical modeling in protocol formulation in cancer chemotherapy. Cancer Treatment Reports 40: 1461–1462. (Reply: Coldman, A.J. and Goldie, J.H. 1986. Cancer Treatment Reports 70: 1461–1462).

    Google Scholar 

  • Sagitov, S. 1989. The limit behavior of reduced critical branching processes. Soviet Mathematics Doklady 38: 488–491.

    MATH  MathSciNet  Google Scholar 

  • Saiki, R.K., Gelfand, D.H., Stoffel, S., Scharf, S.J., Higuchi, R., Horn, G.T., Mullis, K.B. and Erlich, H.A. 1988. Primer-directed enzymatic amplification of DNA with a thermostable DNA polymerase. Science 239: 487–491.

    Google Scholar 

  • Sawyer, S. 1976. Branching diffusion processes in population genetics. Advances in Applied Probability 8: 659–689.

    MATH  MathSciNet  Google Scholar 

  • Seneta, E. and Tavaré, S. 1983. Some stochastic models for plasmid copy number. Theoretical Population Biology 23: 241–256.

    MATH  MathSciNet  Google Scholar 

  • Sennerstam R. 1988. Partition of protein (mass) to sister cell pairs at mitosis: A re-evaluation. Journal of Cell Science 90: 301–306.

    Google Scholar 

  • Sennerstam, R. and Strömberg, J.-O. 1984. A comparative study of the cell cycles of nullipotent and mulitpotent embryonal carcinoma cell lines during exponential growth. Developmental Biology 103: 221–229.

    Google Scholar 

  • Sennerstam R. and Strömberg, J.-O. 1988. Evidence for an intraclonal random shift between two types of cell cycle times in an embryonal carcinoma cell line. Journal of Theoretical Biology 131: 151–162.

    Google Scholar 

  • Sennerstam, R. and Strömberg, J.-O. 1995. Cell cycle progression: Computer simulation of uncoupled subcycles of DNA replication and cell growth. Journal of Theoretical Biology 175: 177–189.

    Google Scholar 

  • Sennerstam, R. and Strömberg, J.-O. 1996. Exponential growth, random transitions and progress through the G1 phase: Computer simulation of experimental data. Cell Proliferation 29: 609–622.

    Google Scholar 

  • Shaw, C.A. 2000. Genealogical methods for multitype branching processes with applications in biology. Ph.D dissertation, Department of Statistics, Rice University, Houston, TX.

    Google Scholar 

  • Shenkar, R., Navidi, W., Tavare, S., Dang, M. H., Chomyn A., Attardi, G., Cortopassi, G., and Arnheim, N. 1996. The mutation rate of the human mtDNA deletion mtDNA4977. American Journal of Human Genetics 59:772–780

    Google Scholar 

  • Spătaru, A. 1989. Properties of branching processes with denumerably many types. Revue Roumaine de Mathématiques Pures et Appliquées (Romanian Journal of Pure and Applied Mathematics) 34: 747–759.

    MATH  Google Scholar 

  • Staiano-Coico, L., Hajjar, D.P., Hefton, J.M., Hajjar, K. and Kimmel, M. 1988. Interaction of arterial cells: III. Stathmokinetic analyses of smooth muscle cells cocultured with endothelial cells. Journal of Cellular Physiology 134: 485–490.

    Google Scholar 

  • Staudte, R.G. 1992. A bifurcating autoregression model for cell lineage data with varying generation means. Journal of Theoretical Biology 156: 183–195.

    Google Scholar 

  • Staudte, R.G., Guiguet, M. and ďHooghe, M.C. 1984. Additive models for dependent cell populations. Journal of Theoretical Biology 109: 127–146.

    Google Scholar 

  • Staudte, R.G., Huggins, R.M., Zhang, J., Axelrod, D.E. and Kimmel, M. 1997. Estimating clonal heterogeneity and interexperiment variability with the bifurcating autoregression model for cell lineage data. Mathematical Biosciences 143: 103–121.

    MATH  Google Scholar 

  • Stewart, F.M., Gordon, D.M. and Levin, B.R. 1990. Fluctuation analysis: The probability distribution of the number of mutants under different conditions. Genetics 124: 175–185.

    Google Scholar 

  • Stigler, S.M. 1970. Estimating the age of a Galton-Watson branching process. Biometrika 57: 505–512.

    MATH  MathSciNet  Google Scholar 

  • Stivers, D.N. and Kimmel, M. 1996a. A continuous-time, multi-type generational inheritance branching process model of cell proliferation with clonal memory. Nonlinear World 3: 385–399.

    MATH  MathSciNet  Google Scholar 

  • Stivers, D.N. and Kimmel, M. 1996b. On the clonal inheritance model of cell proliferation. Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, August 1992. Walter de Gruyter, Berlin, pp. 3401–3408.

    Google Scholar 

  • Stivers, D.N., Kimmel, M. and Axelrod, D.E. 1996. A discrete-time, multi-type generational inheritance branching process model of cell proliferation. Mathematical Biosciences 137: 25–50.

    MATH  Google Scholar 

  • Stoneking, M., Sherry, S.T., Redd, A.J. and Vigilant, L. 1992. New approaches to dating suggest a recent age for the human mtDNA ancestor. Philosophical Transactions of the Royal Society of London B, Biological Sciences 337: 167–175.

    Google Scholar 

  • Swierniak A. and Kimmel, M. 1984. Optimal control application to leukemia chemotherapy protocols design. Technical reports of the Silesian Technical University, Series Automation (Zeszyty Naukowe Politechniki Slaskiej, Seria Automatyka) 73: 261–277 (in Polish, English abstract).

    Google Scholar 

  • Swierniak A. and Kimmel, M. 1991. Cancer cell synchronization and recruitment as optimal control problems. Proceedings of 13th World IMACS Congress, Dublin, Vol. 3, pp. 1461–1462.

    Google Scholar 

  • Swierniak, A. Polanski, A. and Kimmel, M. 1996. Control problems arising in chemotherapy under evolving drug resistance. Preprints of the 13th World Congress of IFAC, Volume B, 411–416.

    Google Scholar 

  • Taïb, Z. 1987. Labelled branching processes with applications to neutral evolution theory. Ph.D. thesis, Chalmers University of Technology, Sweden.

    Google Scholar 

  • Taïb, Z. 1992. Branching Processes and Neutral Evolution. Lecture Notes in Biomathematics, 93. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Taïb, Z. 1993. A note on modeling the dynamics of budding yeast populations using branching process. Journal of Mathematical Biology 31: 805–815.

    MATH  MathSciNet  Google Scholar 

  • Taïb, Z. 1995. Branching processes and functional-differential equations determining steady-size distributions in cell populations. Journal of Applied Probability 32: 1–10.

    MATH  MathSciNet  Google Scholar 

  • Tan, W.Y. 1982. On the distribution theories for the number of mutants in cell populations. SIAM Journal of Applied Mathematics 42: 719–730.

    MATH  Google Scholar 

  • Tan, W.Y. 1983. On the distribution of the number of mutants at the hypoxanthine-quanine phosphoribosyl transferase locus in Chinese hamster ovary cells. Mathematical Biosciences 67: 175–192.

    MATH  Google Scholar 

  • Tannock, I. 1978. Cell kinetics and chemotherapy: A critical review. Cancer Treatment Reports 62: 1117–1133.

    Google Scholar 

  • Tavaré, S. 1980. Time-reversal and age distribution. 1. Discrete Markov case. Journal of Applied Probability 17: 33–46.

    MATH  MathSciNet  Google Scholar 

  • Tavaré, S. 1984. Line-of-descent and genealogical processes, and their applications in population genetics models. Theoretical Population Biology 26: 119–164.

    MATH  MathSciNet  Google Scholar 

  • Therneau, T.M., Solberg, L.A. Jr. and Jenkins, R.B. 1989. Modeling megakaryocyte development as a branching process. Computers and Mathematics with Applications 18: 959–964.

    MathSciNet  Google Scholar 

  • Till, J.E., McCulloch, E.A. and Siminovitch, L. 1964. A stochastic model of stem cell proliferation, based on the growth of spleen colony-forming cells. Proceedings of the National Academy of Sciences USA 51: 29–36.

    Google Scholar 

  • Tltsy, T., Margolin, B.H. and Lum, K. 1989. Differences in the rates of gene amplification in nontumorigenic and tumorigenic cell lines as measured by Luria-Delbrück fluctuation analysis. Proceedings of the National Academy of Sciences USA 86: 9441–9445.

    Google Scholar 

  • Tyrcha, J. 1988. Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle. Journal of Mathematical Biology 26: 465–475.

    MATH  MathSciNet  Google Scholar 

  • Tyson, J.J. 1987. Size control of cell division. Journal of Theoretical Biology 126: 381–391.

    MathSciNet  Google Scholar 

  • Tyson, J.J. and Hannsgen, K.B. 1985a. Global asymptotic stability of the size distribution in probabilistic models of the cell cycle. Journal of Mathematical Biology 22: 61–68.

    MATH  MathSciNet  Google Scholar 

  • Tyson, J.J. and Hannsgen, K.B. 1985b. The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions. Journal of Theoretical Biology 113: 29–62.

    MathSciNet  Google Scholar 

  • Tyson, J.J. and Hannsgen, K.B. 1986. Cell growth and division: A deterministic/probabilistic model of the cell cycle. Journal of Mathematical Biology 23: 231–246.

    MATH  MathSciNet  Google Scholar 

  • Tyson, J., Garcia-Herdugo, G. and Sachsenmaier, W. 1979. Control of nuclear division in Physarum polycephalum. Experimental Cell Research 119: 87–98.

    Google Scholar 

  • Varshaver, N.B., Marshak, M.I. and Shapiro, N.I. 1983. The mutational origin of serum independence in Chinese hamster cells in vitro. International Journal of Cancer 31: 471–475.

    Google Scholar 

  • Venter, J.C. et al. 2001. The sequence of the human genome. Science 291: 1304–1351.

    Google Scholar 

  • Vigilant, L.R., Pennington, H., Harpending, H., Kocher, T.D. and Wilson, A. 1989. Mitochondrial DNA sequences in single hairs from a southern African population. Proceedings of the National Academy of Sciences USA 86: 9350–9354.

    Google Scholar 

  • Vigilant, L., Stoneking, R., Harpending, H., Hawkes, K. and Wilson, A. 1991. African populations and the evolution of human mitochondrial DNA. Science 253: 1503–1507.

    Google Scholar 

  • Vogel, H., Niewisch, H. and Matioli, G. 1969. Stochastic development of stem cells. Journal Theoretical Biology 22: 249–270.

    Google Scholar 

  • Waugh, W.A.O’N. 1981. Application of the Galton-Watson process to the kin number problem. Advances in Applied Probability 13: 631–649.

    MATH  MathSciNet  Google Scholar 

  • Webb, G.F. 1987. Random transitions, size control, and inheritance in cell population dynamics. Mathematical Biosciences 85: 71–91.

    MATH  MathSciNet  Google Scholar 

  • Webb, G.F. 1989. Alpha-and beta-curves, sister-sister and mother-daughter correlations in cell population dynamics. Computers and Mathematics with Applications 18: 973–984.

    MATH  MathSciNet  Google Scholar 

  • Weiss, G. and von Haeseler, A. 1997. A coalescent approach to the polymerase chain reaction. Nucleic Acids Research 25: 3082–3087.

    Google Scholar 

  • Wilson, A.C. and Cann, R.L. 1992. Recent African genesis of humans. Scientific American 266(4): 68–73.

    Google Scholar 

  • Windle, B., Draper, B.W., Yin, Y., O’Gorman, S. and Wahl, G.M. 1991. A central role for chromosome breakage in gene amplification, deletion formation, and amplicon integration. Genes & Development 5: 160–174.

    Google Scholar 

  • Yakovlev, A.Yu. and Yanev, N.M. 1989. Transient Processes in Cell Proliferation Kinetics. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Yule, U.G. 1924. A mathematical theory of evolution based on conclusions of Dr. J.C. Willis, F.R.S. Philosophical Transactions of the Royal Society of London, Series B 213: 21–87.

    Google Scholar 

  • Zakian, V.A. 1995. Telomeres: Beginning to understand the end. Science 270: 1601–1607.

    Google Scholar 

  • Zakian, V.A. 1996. Structure, function, and replication of Saccharomyces cerevisiae telomeres. Annual Review of Genetics 30: 141–172.

    Google Scholar 

  • Zubkov, A.M. 1975. Limiting distribution for the distance to the closest mutual ancestor. Theory of Probability and Its Applications 20: 602–612.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Kimmel, M., Axelrod, D.E. (2002). References. In: Branching Processes in Biology. Interdisciplinary Applied Mathematics, vol 19. Springer, New York, NY. https://doi.org/10.1007/0-387-21639-1_8

Download citation

  • DOI: https://doi.org/10.1007/0-387-21639-1_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-95340-3

  • Online ISBN: 978-0-387-21639-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics