  This paper is concerned with conditions for the existence of oscillations in
gene regulatory networks with negative cyclic feedback, where time delays in
transcription, translation and translocation process are explicitly considered.
The primary goal of this paper is to propose systematic analysis tools that are
useful for a broad class of cyclic gene regulatory networks, and to provide
novel biological insights. To this end, we adopt a simplified model that is
suitable for capturing the essence of a large class of gene regulatory
networks. It is first shown that local instability of the unique equilibrium
state results in oscillations based on a Poincare-Bendixson type theorem. Then,
a graphical existence condition, which is equivalent to the local instability
of a unique equilibrium, is derived. Based on the graphical condition, the
existence condition is analytically presented in terms of biochemical
parameters. This allows us to find the dimensionless parameters that primarily
affect the existence of oscillations, and to provide biological insights. The
analytic conditions and biological insights are illustrated with two existing
biochemical networks, Repressilator and the Hes7 gene regulatory networks.
