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The Fractional Birth Process with Power-Law Immigration

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Abstract

A semi-stochastic description of the mutation process with memory effects in a power-law growing cell colony is considered. Specifically, we give explicit expressions for the distribution of the number of mutants in a single clone, and of the total number of mutants. The investigation is performed under the assumption that clones grow according to a fractional linear birth process, characterized by a non-exponential, Mittag-Leffler waiting time distribution. Its slowly decaying long tail enables modeling bursty dynamics: very dense sequences of events are separated by long times of reduced activity. The probabilistic construction also allows for recovering the mean and the variance of the total number of mutants. We then give exact formulas for the higher-order moments of the fractional linear birth process and of the clone size, thus providing additional insight into this evolutionary process.

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References

  1. Altrock, P.M., Liu, L.L., Michor, F.: The mathematics of cancer: integrating quantitative models. Nat. Rev. Cancer 15(12), 730 (2015)

    Article  Google Scholar 

  2. Barabasi, A.L.: The origin of bursts and heavy tails in human dynamics. Nature 435(7039), 207 (2005)

    Article  ADS  Google Scholar 

  3. Beghin, L., Orsingher, E.: Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15, 684–709 (2010)

    Article  MathSciNet  Google Scholar 

  4. Benzekry, S., Lamont, C., Beheshti, A., Tracz, A., Ebos, J.M., Hlatky, L., Hahnfeldt, P.: Classical mathematical models for description and prediction of experimental tumor growth. PLoS Comput. Biol. 10(8), e1003800 (2014)

    Article  ADS  Google Scholar 

  5. Cahoy, D.O., Polito, F.: Simulation and estimation for the fractional Yule process. Methodol. Comput. Appl. Probab. 14(2), 383–403 (2012)

    Article  MathSciNet  Google Scholar 

  6. Cahoy, D.O., Polito, F.: Parameter estimation for fractional birth and fractional death processes. Stat. Comput. 24(2), 211–222 (2014)

    Article  MathSciNet  Google Scholar 

  7. Cahoy, D.O., Uchaikin, V.V., Woyczynski, W.A.: Parameter estimation for fractional Poisson processes. J. Stat. Plan. Infer. 140(11), 3106–3120 (2010)

    Article  MathSciNet  Google Scholar 

  8. Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics, vol. 378. Springer, New York (2014)

    MATH  Google Scholar 

  9. Di Crescenzo, A., Martinucci, B., Meoli, A.: A fractional counting process and its connection with the Poisson process. ALEA 13(1), 291–307 (2016)

    Article  MathSciNet  Google Scholar 

  10. Feng, C., Pettersson, M., Lamichhaney, S., Rubin, C.J., Rafati, N., Casini, M., Folkvord, A., Andersson, L.: Moderate nucleotide diversity in the atlantic herring is associated with a low mutation rate. Elife 6, e23907 (2017)

    Article  Google Scholar 

  11. Garra, R., Garrappa, R.: The Prabhakar or three parameter Mittag-Leffler function: theory and application. Commun. Nonlinear Sci. Numer. Simul. 56, 314–329 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  12. Gorenflo, R., Mainardi, F.: The asymptotic universality of the Mittag-Leffler waiting time law in continuous random walks. In: Lecture note at WE-Heraeus-Seminar on Physikzentrum Bad-Honnef (Germany) pp. 12–16 (2006)

  13. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, New York (2014)

    Book  Google Scholar 

  14. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Longman Publishing Co. Inc, Boston (1994)

    MATH  Google Scholar 

  15. Jafarpour, F., Wright, C.S., Gudjonson, H., Riebling, J., Dawson, E., Lo, K., Fiebig, A., Crosson, S., Dinner, A.R., Iyer-Biswas, S.: Bridging the timescales of single-cell and population dynamics. Phys. Rev. X 8(2), 021007 (2018)

    Google Scholar 

  16. Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109(3), 217–234 (2002)

    MATH  Google Scholar 

  17. Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Elsevier, Amsterdam (1981)

    MATH  Google Scholar 

  18. Keller, P., Antal, T.: Mutant number distribution in an exponentially growing population. J. Stat. Mech. 2015(1), P01011 (2015)

    Article  MathSciNet  Google Scholar 

  19. Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Amsterdam (2006)

    MATH  Google Scholar 

  20. Klein, A.M., Brash, D.E., Jones, P.H., Simons, B.D.: Stochastic fate of p53-mutant epidermal progenitor cells is tilted toward proliferation by UV B during preneoplasia. Proc. Natl. Acad. Sci. 107(1), 270–275 (2010)

    Article  ADS  Google Scholar 

  21. Luria, S.E., Delbrück, M.: Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28(6), 491 (1943)

    Google Scholar 

  22. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7(9), 1461–1477 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  23. Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–38 (1996)

    Article  MathSciNet  Google Scholar 

  24. Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer, New York (2008)

    Book  Google Scholar 

  25. Melzak, Z.A., Newman, D.J., Erdos, P., Grossman, G., Spiegel, M.R.: Problems for solution: 4458–4462. Am. Math. Mon. 58(9), 636–636 (1951)

    Article  Google Scholar 

  26. Nicholson, M.D., Antal, T.: Universal asymptotic clone size distribution for general population growth. Bull. Math. Biol. 78(11), 2243–2276 (2016)

    Article  MathSciNet  Google Scholar 

  27. Orsingher, E., Polito, F.: Fractional pure birth processes. Bernoulli 16(3), 858–881 (2010)

    Article  MathSciNet  Google Scholar 

  28. Ortigueira, M., Bengochea, G.: A new look at the fractionalization of the logistic equation. Physica A 467, 554–561 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  29. Quaintance, J.: Combinatorial Identities for Stirling Numbers: The Unpublished Notes of HW Gould. World Scientific, Singapore (2015)

    Book  Google Scholar 

  30. Rosenfeld, S.: Origins of stochasticity and burstiness in high-dimensional biochemical networks. EURASIP J. Bioinform. Syst. Biol. 2009(1), 362309 (2008)

    Google Scholar 

  31. Solomonovich, I.A., Zwillinger, D., et al.: Table of Integrals, Series, and Products. Elsevier, Amsterdam (2014)

    Google Scholar 

  32. Uchaikin, V.V., Cahoy, D.O., Sibatov, R.T.: Fractional processes: from Poisson to branching one. Int. J. Bifurcat. Chaos 18(09), 2717–2725 (2008)

    Article  MathSciNet  Google Scholar 

  33. Zheng, Q.: Progress of a half century in the study of the Luria-Delbrück distribution. Math. Biosci. 162(1–2), 1–32 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank Tyll Krüger and Viktor Bezborodov for many insightful discussions, and Antonio Di Crescenzo for helpful comments which improved the presentation of the paper. Special thanks are due to Giacomo Ascione for technical discussions and help. Last but not least, many thanks to the referees for the careful reading of the manuscript and for providing many corrections and suggestions. This paper has been performed under partial support by MIUR (PRIN 2017, project “Stochastic Models for Complex Systems” no. 2017JFFHSH). A.M. is member of the Research group GNCS of INdAM.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Alessandra Meoli

  2. Department of Biosystems Science and Engineering, ETH Zürich, Mattenstrasse 26, 4058, Basel, Switzerland

    Niko Beerenwinkel & Mykola Lebid

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  1. Alessandra Meoli
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Correspondence to Alessandra Meoli.

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Communicated by Abhishek Dhar.

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Meoli, A., Beerenwinkel, N. & Lebid, M. The Fractional Birth Process with Power-Law Immigration. J Stat Phys 178, 775–799 (2020). https://doi.org/10.1007/s10955-019-02455-5

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