Quasistationarity in a Branching Model of Division-Within-Division

Abstract

Many biological processes include branching phenomena, which may be called division-within-division. Examples are gene amplification in cancer cells and elsewhere, plasmid dynamics in bacteria, and proliferation of viral particles in host cells. In some cases, the loss of “smaller” particles from the “large” ones leads to extinction of the latter. The logical question is then to ask about the distribution of the nonextinct particles, which mathematically leads to the consideration of quasistationarity, ie. stationarity of the process conditional on nonabsorption.

We consider a model in which the large particles follow a supercritical process, while the small ones divide subcritically. We demonstrate that the part of population of the large particles which contain at least 1 small particle may expand or decay, and that the distribution of the number of small particles in large particles tends to a limit. We also discuss biological significance of results of this type.

Articlefootnote

The author was supported by the NSF grants DMS 9203436 and DMS 9409909 and by the Keck’s Center for Computational Biology.