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

TL;DR

This paper models the vertical transfer of plasmids in bacteria using a hyperbolic integro-differential equation, analyzes its spectral properties, and provides methods for numerical eigenfunction construction to understand long-term plasmid distribution.

Contribution

It introduces a continuous growth-fragmentation-death model for plasmid dynamics and analyzes its spectral properties, including existence and stability of eigensolutions.

Findings

Existence of a principal eigenvalue and eigenfunction for the model.

Spectral analysis reveals stability conditions for plasmid distribution.

An iterative numerical method for eigenfunction approximation is developed.

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.