  RNA interference (RNAi) is a fundamental cellular process that inhibits gene
expression through cleavage and destruction of target mRNA. It is responsible
for a number of important intracellular functions, from being the first line of
immune defence against pathogens to regulating development and morphogenesis.
In this paper we consider a mathematical model of RNAi with particular emphasis
on time delays associated with two aspects of primed amplification: binding of
siRNA to aberrant RNA, and binding of siRNA to mRNA, both of which result in
the expanded production of dsRNA responsible for RNA silencing. Analytical and
numerical stability analyses are performed to identify regions of stability of
different steady states and to determine conditions on parameters that lead to
instability. Our results suggest that while the original model without time
delays exhibits a bi-stability due to the presence of a hysteresis loop, under
the influence of time delays, one of the two steady states with the high
(default) or small (silenced) concentration of mRNA can actually lose its
stability via a Hopf bifurcation. This leads to the co-existence of a stable
steady state and a stable periodic orbit, which has a profound effect on the
dynamics of the system.
