  Gene regulatory circuits show significant stochastic fluctuations in their
circuit signals due to the low copy number of transcription factors. When a
gene circuit component is connected to an existing circuit, the dynamic
properties of the existing circuit can be affected by the connected component.
In this paper, we investigate modularity in the dynamics of the gene circuit
based on stochastic fluctuations in the circuit signals. We show that the noise
in the output signal of the existing circuit can be affected significantly when
the output is connected to the input of another circuit component. More
specifically, the output signal noise can show significantly longer
correlations when the two components are connected. This equivalently means
that the noise power spectral density becomes narrower. We define the relative
change in the correlation time or the spectrum bandwidth by stochastic
retroactivity, which is shown to be directly related to the retroactivity
defined in the deterministic framework by del Vecchio et al. This provides an
insight on how to measure retroactivity, by investigating stochastic
fluctuations in gene expression levels, more specifically, by obtaining an
autocorrelation function of the fluctuations. We also provide an interesting
aspect of the frequency response of the circuit. We show that depending on the
magnitude of operating frequencies, different kinds of signals need to be
preferably chosen for circuit description in a modular fashion: at low enough
frequency, expression level of transcription factor that are not bound to their
specific promoter region needs to be chosen, and at high enough frequency, that
of the total transcription factor, both bound and unbound, does.

Quantitative Methods