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.
Highlights
Plasmids are mobile genetic elements in bacteria
Plasmids have been studied intensively due to, e.g., their role in the spread of antibiotic resistance genes in bacterial populations [10, 3] and their importance in biotechnology where they are used as vectors [11]
We focus on high-copy plasmids as they are commonly used in biotechnology
Summary
Plasmids are mobile genetic elements in bacteria. They replicate autonomously, and are heritable [10]. Models of a cellular population structured by a continuous variable often assume the form of aggregationfragmentation or growth-fragmentation equations and have been studied extensively [8, 27] These equations are typically analyzed using the theory of semigroups [38, 31], the Laplace transform [20], or theory of positive operators together with compactness [20, 14]. Adapt it to the case of a bounded plasmid number and a plasmid reproduction rate that vanishes for plasmid-free bacteria and bacteria with the maximal plasmid content This method uses a Lyapunov functional to obtain stability results and does not require compactness. Appendix A contains the proof of Theorem 2.9 which is central in the proof of the existence of an eigensolution
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