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Journal of Mathematical Biology

, Volume 78, Issue 5, pp 1299–1330 | Cite as

Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids

  • Eva StadlerEmail author
Article

Abstract

Plasmids are autonomously replicating genetic elements in bacteria. At cell division, plasmids are distributed among the two daughter cells. This gene transfer from one generation to the next is called vertical gene transfer. We study the dynamics of a bacterial population carrying plasmids and are in particular interested in the long-time distribution of plasmids. Starting with a model for a bacterial population structured by the discrete number of plasmids, we proceed to the continuum limit in order to derive a continuous model. The model incorporates plasmid reproduction, division and death of bacteria, and distribution of plasmids at cell division. It is a hyperbolic integro-differential equation and a so-called growth-fragmentation-death model. As we are interested in the long-time distribution of plasmids we study the associated eigenproblem and show existence of eigensolutions. The stability of this solution is studied by analyzing the spectrum of the integro-differential operator given by the eigenproblem. By relating the spectrum with the spectrum of an integral operator we find a simple real dominating eigenvalue with a non-negative corresponding eigenfunction. Moreover, we describe an iterative method for the numerical construction of the eigenfunction.

Keywords

Growth-fragmentation-death equation Plasmid dynamics Hyperbolic PDE Eigenproblem Spectral analysis 

Mathematics Subject Classification

92D25 35L99 35Q80 47A10 47B65 

Notes

Acknowledgements

I want to thank Johannes Müller for intensive discussions. I also want to thank an anonymous reviewer for suggesting various improvements of the manuscript. This work was funded by the German Research Foundation (DFG) priority program SPP1617 “Phenotypic heterogeneity and sociobiology of bacterial populations” (DFG MU 2339/2-2).

References

  1. Arino O (1995) A survey of structured cell population dynamics. Acta Biotheor 43:3–25CrossRefGoogle Scholar
  2. Beebee TJC, Rowe G (2008) An introduction to molecular ecology. Oxford University Press, OxfordGoogle Scholar
  3. Bentley WE, Quiroga OE (1993) Investigation of subpopulation heterogeneity and plasmid stability in recombinant Escherichia coli via a simple segregated model. Biotechnol Bioeng 42(2):222–234CrossRefGoogle Scholar
  4. Bentley WE, Mirjalili N, Andersen DC, Davis RH, Kompala DS (1990) Plasmid-encoded protein: the principal factor in the “Metabolic Burden” associated with recombinant bacteria. Biotechnol Bioeng 35(7):668–681CrossRefGoogle Scholar
  5. Bonsall FF (1955) Endomorphisms of a partially ordered vector space without. Order Unit J Lond Math Soc s1 30(2):144–153MathSciNetCrossRefzbMATHGoogle Scholar
  6. Brezis H (2010) Functional analysis, sobolev spaces and partial differential equations. Springer, New YorkCrossRefGoogle Scholar
  7. Calsina À, Saldaña J (1995) A model of physiologically structured population dynamics with a nonlinear individual growth rate. J Math Biol 33(4):335–364MathSciNetCrossRefzbMATHGoogle Scholar
  8. Calvez V, Doumic-Jauffret M, Gabriel P (2012) Self-similarity in a general aggregation-fragmentation problem; application to fitness analysis. J Math Pures Appl 98(1):1–27MathSciNetCrossRefzbMATHGoogle Scholar
  9. Campillo F, Champagnat N, Fritsch C (2016) Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models. J Math Biol 73(6–7):1781–1821MathSciNetCrossRefzbMATHGoogle Scholar
  10. Casali N, Preston A (eds) (2003) E. coli plasmid vectors: methods and applications. Humana Press, Totowa, NJGoogle Scholar
  11. Cheung YK, Leung AYT (2004) Finite element methods in dynamics, solid mechanics and its application, vol 5. Kluwer, DordrechtGoogle Scholar
  12. Clark DP, Pazdernik NJ (2016) Biotechnology, 2nd edn. Elsevier AP Cell Press, AmsterdamGoogle Scholar
  13. Conway JB (1985) A course in functional analysis, graduate texts in mathematics, vol 96. Springer, New YorkCrossRefGoogle Scholar
  14. Cushing JM (1998) An introduction to structured population dynamics. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  15. Dautray R, Lions JL (1990) Mathematical analysis and numerical methods for science and technology: volume 3 spectral theory and applications. Springer, BerlinGoogle Scholar
  16. Degla G (2008) An overview of semi-continuity results on the spectral radius and positivity. J Math Anal Appl 338:101–110MathSciNetCrossRefzbMATHGoogle Scholar
  17. Doumic M (2007) Analysis of a population model structured by the cells molecular content. Math Model Nat Phenom 2(3):121–152MathSciNetCrossRefzbMATHGoogle Scholar
  18. Doumic-Jauffret M, Gabriel P (2010) Eigenelements of a general aggregation-fragmentation model. Math Models Methods Appl Sci 20(5):757–783MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ganusov VV, Bril’kov AV, Pechurkin NS (2000) Mathematical modeling of population dynamics of unstable plasmid-bearing bacterial strains during continuous cultivation in the chemostat. Biofizika 45(5):908–914Google Scholar
  20. Golub GH, Van Loan CF (2013) Matrix computations, 4th edn, Johns Hopkins studies in the mathematical sciences, vol 3. JHU Press, BaltimoreGoogle Scholar
  21. Heijmans H (1986) The dynamical behaviour of the age-size-distribution of a cell population. In: Metz JAJ, Diekmann O (eds) The dynamics of physiologically structured populations, lecture notes in biomathematics, vol 68. Berlin, Heidelberg, pp 185–202CrossRefGoogle Scholar
  22. Magal P, Ruan S (eds) (2008) Structured population models in biology and epidemiology, lecture notes in mathematics, vol 1936. Springer, BerlinzbMATHGoogle Scholar
  23. Metz JAJ, Diekmann O (eds) (1986) The dynamics of physiologically structured populations, lecture notes in biomathematics, vol 68. Springer, BerlinGoogle Scholar
  24. Michel P (2006) Existence of a solution to the cell division eigenproblem. Math Models Methods Appl Sci 16(supp01):1125–1153MathSciNetCrossRefzbMATHGoogle Scholar
  25. Million-Weaver S, Camps M (2014) Mechanisms of plasmid segregation: have multicopy plasmids been overlooked? Plasmid 75:27–36CrossRefGoogle Scholar
  26. Mischler S, Scher J (2016) Spectral analysis of semigroups and growth-fragmentation equations. Ann Inst H Poincaré Anal Non Linéaire 33(3):849–898MathSciNetCrossRefzbMATHGoogle Scholar
  27. Müller G, Noack D, Schorcht R, Gáspár S, Herényi L (1982) Mathematical modelling of segregation processes in microbial populations containing a single plasmid species. Acta Phys Acad Sci Hung 53(1–2):255–262CrossRefGoogle Scholar
  28. Müller J, Münch K, Koopmann B, Stadler E, Roselius L, Jahn D, Münch R (2017) Plasmid segregation and accumulation. ArXiv e-prints arXiv:1701.03448
  29. Perthame B (2007) Transport equation in biology. Frontiers in mathematics. Birkhäuser, BaselzbMATHGoogle Scholar
  30. Pogliano J, Ho TQ, Zhong Z, Helinski DR (2001) Multicopy plasmids are clustered and localized in Escherichia coli. Proc Natl Acad Sci USA 98(8):4486–4491CrossRefGoogle Scholar
  31. R Core Team (2017) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaGoogle Scholar
  32. Rudin W (1986) Real and complex analysis, 3rd edn. McGraw-Hill International Editions Mathematics Series. McGraw-Hill, New YorkGoogle Scholar
  33. Sawashima I (1964) On spectral properties of some positive operators. Natur Sci Rep Ochanomizu Univ 15(2):53–64MathSciNetzbMATHGoogle Scholar
  34. Stewart FM, Levin BR (1977) The population biology of bacterial plasmids: a priori conditions for the existence of conjugationally transmitted factors. Genetics 87(2):209–228MathSciNetGoogle Scholar
  35. Summers DK (1996) The biology of plasmids. Wiley, OxfordCrossRefGoogle Scholar
  36. Webb GF (1985) Theory of nonlinear age-dependent population dynamics, monographs and textbooks in pure and applied mathematics, vol 89. Marcel Dekker, New YorkGoogle Scholar
  37. Webb GF (2008) Population models structured by age, size, and spatial position. In: Magal P, Ruan S (eds) Structured population models in biology and epidemiology, lecture notes in mathematics, vol 1936, Springer, BerlinGoogle Scholar
  38. Yosida K (1995) Functional analysis, classics in mathematics, vol 123. Springer, BerlinCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität MünchenGarching, MunichGermany

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