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

Eva Stadler; Johannes Müller

doi:10.3934/dcdsb.2020091

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2020, Volume 25, Issue 11: 4127-4164. Doi: 10.3934/dcdsb.2020091

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

Eva Stadler^1,2, , and
Johannes Müller^1,3,

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
^* Corresponding author: Eva Stadler

Received: May 2019

Revised: November 2019

Early access: April 2020

Published: November 2020

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).

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Abstract

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.

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Mathematics Subject Classification: 92D25, 35Q92, 35L02, 35B40, 44A10.

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Figure 1. 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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