# American Institute of Mathematical Sciences

November  2020, 25(11): 4127-4164. doi: 10.3934/dcdsb.2020091

## Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case

 1 Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany 2 Present address: Infection Analytics Program, Kirby Institute, UNSW Sydney, Wallace Wurth Building, High St, Kensington NSW 2052, Australia 2 Institute of Computational Biology, HelmholtzZentrum München - German Research Center for, Environmental Health, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany

* Corresponding author: Eva Stadler

Received  May 2019 Revised  November 2019 Published  November 2020 Early access  April 2020

Fund Project: This work is part of the published dissertation thesis "Transport equations and plasmid-induced cellular heterogeneity" by ES (https://mediatum.ub.tum.de/1469742?id=1469742).
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).

We study the distribution of autonomously replicating genetic elements, so-called plasmids, in a bacterial population. When a bacterium divides, the plasmids are segregated between the two daughter cells. We analyze a model for a bacterial population structured by their plasmid content. The model contains reproduction of both plasmids and bacteria, death of bacteria, and the distribution of plasmids at cell division. The model equation is a growth-fragmentation-death equation with an integral term containing a singular kernel. As we are interested in the long-term distribution of the plasmids, we consider the associated eigenproblem. Due to the singularity of the integral kernel, we do not have compactness. Thus, standard approaches to show the existence of an eigensolution like the Theorem of Krein-Rutman cannot be applied. We show the existence of an eigensolution using a fixed point theorem and the Laplace transform. The long-term dynamics of the model is analyzed using the Generalized Relative Entropy method.

Citation: Eva Stadler, Johannes Müller. Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4127-4164. doi: 10.3934/dcdsb.2020091
##### References:
 [1] O. Arino, A survey of structured cell population dynamics, Acta Biotheor., 43 (1995), 3-25.  doi: 10.1007/BF00709430. [2] K. B. Athreya and P. E. Ney, Branching Processes, Die Grundlehren der mathematischen Wissenschaften, vol. 196, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65371-1. [3] T. Beebee and G. Rowe, An Introduction to Molecular Ecology, 2nd edition, Oxford University Press, Oxford University Press, 2008. [4] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rend. Circ. Mat. Palermo (2), 7 (1958), 261-269. [5] W. E. Bentley, N. Mirjalili, D. C. Andersen, R. H. Davis and D. S. Kompala, Plasmid-encoded protein: The principal factor in the "metabolic burden" associated with recombinant bacteria, Biotechnol. Bioeng., 35 (1990), 668-681.  doi: 10.1002/bit.260350704. [6] V. I. Bogachev, Measure Theory, vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [7] À. Calsina and J. Saldaña, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377. [8] V. Calvez, M. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem: Application to fitness analysis, J. Math. Pures Appl., 98 (2012), 1-27.  doi: 10.1016/j.matpur.2012.01.004. [9] F. Campillo, N. Champagnat and C. Fritsch, Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models, J. Math. Biol., 73 (2016), 1781-1821.  doi: 10.1007/s00285-016-1012-6. [10] N. Casali and A. Preston, E. coli Plasmid Vectors: Methods and Applications, Methods in Molecular Biology, vol. 235, Humana Press, Totowa, NJ, 2003. doi: 10.1385/1592594093. [11] D. P. Clark and N. J. Pazdernik, Biotechnology, 2$^nd$ edition, Elsevier AP Cell Press, Amsterdam, 2015. [12] J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [13] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-3-642-65690-3. [14] M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006. [15] M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.  doi: 10.1142/S021820251000443X. [16] S. M. Focardi and F. J. Fabozzi, The Mathematics of Financial Modeling and Investment Management, 1$^st$ edition, Frank J. Fabozzi Series, John Wiley & Sons, 2004. [17] V. V. Ganusov, A. V. Bril'kov and N. S. Pechurkin, Mathematical modeling of population dynamics of unstable plasmid-bearing bacteria strains during continuous cultivation in a chemostat, Biofizika, 45 (2000), 908-914. [18] L. M. Graves, The Theory of Functions of Real Variables, 2$^nd$ edition, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. [19] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 3$^rd$ edition, Graduate studies in mathematics, vol. 40, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/gsm/040. [20] H. J. A. M. Heijmans, The dynamical behaviour of the age-size-distribution of a dell population, in The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath., vol. 68, Springer, Berlin, 1986,185–202. doi: 10.1007/978-3-662-13159-6_5. [21] H. Kuo and J. D. Keasling, A Monte Carlo simulation of plasmid replication during the bacterial division cycle, Biotechnol. Bioeng., 52 (1996), 633-647.  doi: 10.1002/(SICI)1097-0290(19961220)52:6<633::AID-BIT1>3.0.CO;2-P. [22] P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5. [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68, Springer, Berlin, Heidelberg, 1986. [24] P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153.  doi: 10.1142/S0218202506001480. [25] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001. [26] S. Million-Weaver and M. Camps, Mechanisms of plasmid segregation: Have multicopy plasmids been overlooked?, Plasmid, 75 (2014), 27-36.  doi: 10.1016/j.plasmid.2014.07.002. [27] S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007. [28] J. Müller, K. Münch, B. Koopmann, E. Stadler, L. Roselius, D. Jahn and R. Münch, Plasmid segregation and accumulation, preprint, arXiv: 1701.03448. [29] K. M. Münch, J. Müller, S. Wienecke, S. Bergmann, S. Heyber, R. Biedendieck and et al., Polar fixation of plasmids during recombinant protein production in bacillus megaterium results in population heterogeneity, Applied Environmental Microbiology, 81 (2015), 5976-5986.  doi: 10.1128/AEM.00807-15. [30] R. P. Novick, R. C. Clowes, S. N. Cohen, R. Curtiss, N. Datta and S. Falkow, Uniform nomenclature for bacterial plasmids: A proposal, Microbiology and Molecular Biology Reviews, 40 (1976), 168-189.  doi: 10.1128/MMBR.40.1.168-189.1976. [31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [32] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4. [33] J. Pogliano, T. Q. Ho, Z. Zhong and D. R. Helinski, Multicopy plasmids are clustered and localized in Escherichia coli, PNAS, 98 (2001), 4486-4491.  doi: 10.1073/pnas.081075798. [34] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2018. [35] S. Srivastava, Genetics of Bacteria, Springer, India, 2013. doi: 10.1007/978-81-322-1090-0. [36] E. Stadler, Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids, J. Math. Biol., 78 (2019), 1299-1330.  doi: 10.1007/s00285-018-1310-2. [37] F. M. Stewart and B. R. Levin, The population biology of bacterial plasmids: A priori conditions for the existence of conjugationally transmitted factors, Genetics, 87 (1977), 209-228. [38] G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936, Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1. [39] D. G. Zill and P. Shanahan, A First Course in Complex Analysis with Applications, Jones and Bartlett, Boston, 2003.

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##### References:
 [1] O. Arino, A survey of structured cell population dynamics, Acta Biotheor., 43 (1995), 3-25.  doi: 10.1007/BF00709430. [2] K. B. Athreya and P. E. Ney, Branching Processes, Die Grundlehren der mathematischen Wissenschaften, vol. 196, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65371-1. [3] T. Beebee and G. Rowe, An Introduction to Molecular Ecology, 2nd edition, Oxford University Press, Oxford University Press, 2008. [4] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rend. Circ. Mat. Palermo (2), 7 (1958), 261-269. [5] W. E. Bentley, N. Mirjalili, D. C. Andersen, R. H. Davis and D. S. Kompala, Plasmid-encoded protein: The principal factor in the "metabolic burden" associated with recombinant bacteria, Biotechnol. Bioeng., 35 (1990), 668-681.  doi: 10.1002/bit.260350704. [6] V. I. Bogachev, Measure Theory, vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [7] À. Calsina and J. Saldaña, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377. [8] V. Calvez, M. Doumic and P. Gabriel, Self-similarity in a general aggregation-fragmentation problem: Application to fitness analysis, J. Math. Pures Appl., 98 (2012), 1-27.  doi: 10.1016/j.matpur.2012.01.004. [9] F. Campillo, N. Champagnat and C. Fritsch, Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models, J. Math. Biol., 73 (2016), 1781-1821.  doi: 10.1007/s00285-016-1012-6. [10] N. Casali and A. Preston, E. coli Plasmid Vectors: Methods and Applications, Methods in Molecular Biology, vol. 235, Humana Press, Totowa, NJ, 2003. doi: 10.1385/1592594093. [11] D. P. Clark and N. J. Pazdernik, Biotechnology, 2$^nd$ edition, Elsevier AP Cell Press, Amsterdam, 2015. [12] J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [13] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-3-642-65690-3. [14] M. Doumic, Analysis of a population model structured by the cells molecular content, Math. Model. Nat. Phenom., 2 (2007), 121-152.  doi: 10.1051/mmnp:2007006. [15] M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.  doi: 10.1142/S021820251000443X. [16] S. M. Focardi and F. J. Fabozzi, The Mathematics of Financial Modeling and Investment Management, 1$^st$ edition, Frank J. Fabozzi Series, John Wiley & Sons, 2004. [17] V. V. Ganusov, A. V. Bril'kov and N. S. Pechurkin, Mathematical modeling of population dynamics of unstable plasmid-bearing bacteria strains during continuous cultivation in a chemostat, Biofizika, 45 (2000), 908-914. [18] L. M. Graves, The Theory of Functions of Real Variables, 2$^nd$ edition, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. [19] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 3$^rd$ edition, Graduate studies in mathematics, vol. 40, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/gsm/040. [20] H. J. A. M. Heijmans, The dynamical behaviour of the age-size-distribution of a dell population, in The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath., vol. 68, Springer, Berlin, 1986,185–202. doi: 10.1007/978-3-662-13159-6_5. [21] H. Kuo and J. D. Keasling, A Monte Carlo simulation of plasmid replication during the bacterial division cycle, Biotechnol. Bioeng., 52 (1996), 633-647.  doi: 10.1002/(SICI)1097-0290(19961220)52:6<633::AID-BIT1>3.0.CO;2-P. [22] P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5. [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68, Springer, Berlin, Heidelberg, 1986. [24] P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153.  doi: 10.1142/S0218202506001480. [25] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260.  doi: 10.1016/j.matpur.2005.04.001. [26] S. Million-Weaver and M. Camps, Mechanisms of plasmid segregation: Have multicopy plasmids been overlooked?, Plasmid, 75 (2014), 27-36.  doi: 10.1016/j.plasmid.2014.07.002. [27] S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007. [28] J. Müller, K. Münch, B. Koopmann, E. Stadler, L. Roselius, D. Jahn and R. Münch, Plasmid segregation and accumulation, preprint, arXiv: 1701.03448. [29] K. M. Münch, J. Müller, S. Wienecke, S. Bergmann, S. Heyber, R. Biedendieck and et al., Polar fixation of plasmids during recombinant protein production in bacillus megaterium results in population heterogeneity, Applied Environmental Microbiology, 81 (2015), 5976-5986.  doi: 10.1128/AEM.00807-15. [30] R. P. Novick, R. C. Clowes, S. N. Cohen, R. Curtiss, N. Datta and S. Falkow, Uniform nomenclature for bacterial plasmids: A proposal, Microbiology and Molecular Biology Reviews, 40 (1976), 168-189.  doi: 10.1128/MMBR.40.1.168-189.1976. [31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [32] B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. doi: 10.1007/978-3-7643-7842-4. [33] J. Pogliano, T. Q. Ho, Z. Zhong and D. R. Helinski, Multicopy plasmids are clustered and localized in Escherichia coli, PNAS, 98 (2001), 4486-4491.  doi: 10.1073/pnas.081075798. [34] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2018. [35] S. Srivastava, Genetics of Bacteria, Springer, India, 2013. doi: 10.1007/978-81-322-1090-0. [36] E. Stadler, Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids, J. Math. Biol., 78 (2019), 1299-1330.  doi: 10.1007/s00285-018-1310-2. [37] F. M. Stewart and B. R. Levin, The population biology of bacterial plasmids: A priori conditions for the existence of conjugationally transmitted factors, Genetics, 87 (1977), 209-228. [38] G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936, Springer, Berlin, 2008, 1–49. doi: 10.1007/978-3-540-78273-5_1. [39] D. G. Zill and P. Shanahan, A First Course in Complex Analysis with Applications, Jones and Bartlett, Boston, 2003.
Numerically constructed eigenfunctions for $\Phi(\xi) = 6\,\xi\,(1-\xi)$, $\mu = 0.1/h$, $b(z) = z(1-z)/h$, and different $\beta$, viz. $\beta = 0.45/h$ (black), $0.5/h$ (dark gray), and $0.55/h$ (light gray). The different cell division rates lead to different behavior of the eigenfunction $\mathcal{U}(z)$ at the maximal plasmid number $z_0 = 1$. The eigenfunction was numerically constructed using the software R [34] as described in [36,Section 5]
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