  Homeostasis of protein concentrations in cells is crucial for their proper
functioning, and this requires concentrations (at their steady-state levels) to
be stable to fluctuations. Since gene expression is regulated by proteins such
as transcription factors (TFs), the full set of proteins within the cell
constitutes a large system of interacting components. Here, we explore factors
affecting the stability of this system by coupling the dynamics of mRNAs and
protein concentrations in a growing cell. We find that it is possible for
protein concentrations to become unstable if the regulation strengths or system
size becomes too large, and that other global structural features of the
networks can dramatically enhance the stability of the system. In particular,
given the same number of proteins, TFs, number of interactions, and regulation
strengths, a network that resembles a bipartite graph with a lower fraction of
interactions that target TFs has a higher chance of being stable. By scrambling
the $\textit{E. coli.}$ transcription network, we find that the randomized
network with the same number of regulatory interactions is much more likely to
be unstable than the real network. These findings suggest that constraints
imposed by system stability could have played a role in shaping the existing
regulatory network during the evolutionary process. We also find that contrary
to what one might expect from random matrix theory and what has been argued in
the literature, the degradation rate of mRNA does not affect whether the system
is stable.
