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

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.