Asymptotic Analysis of a Cell Cycle Model Based on Unequal Division

We provide an analysis of the asymptotic behavior of a novel cell cycle model offering a uniform description of the processes of RNA production and division and explaining the cell generation time variability, at least in certain cell lines. We prove that the distribution $m( t,x )$ of the RNA level $( x )$ in the dividing cells at a given time $( t )$ tends to $\exp ( \lambda^* t )\mu^* ( x )$, as t tends to infinity, i.e., that there exists the so-called exponential steady state for our model, and that perturbed cell populations reach this state in the limit. The result is obtained by applying the semigroup theory to a functional integral equation describing the evolution of $m( t,x )$ in time. For a variety of reasons, the analysis requires explicit characterization of the semigroup spectrum. Our asymptotic result may be viewed as a generalization of similar results for the generalized branching processes with continuous time, under less restrictive assumptions. Also, we discuss related models and asymptotic results including the results obtained using the notion of essential spectrum of an operator. The present paper is a continuation of another work in which the biological background and discussions were provided.