  Gene products (RNAs, proteins) often occur at low molecular counts inside
individual cells, and hence are subject to considerable random fluctuations
(noise) in copy number over time. Not surprisingly, cells encode diverse
regulatory mechanisms to buffer noise. One such mechanism is the incoherent
feedforward circuit. We analyze a simplistic version of this circuit, where an
upstream regulator X affects both the production and degradation of a protein
Y. Thus, any random increase in X's copy numbers would increase both production
and degradation, keeping Y levels unchanged. To study its stochastic dynamics,
we formulate this network into a mathematical model using the Chemical Master
Equation formulation. We prove that if the functional dependence of Y's
production and degradation on X is similar, then the steady-distribution of Y's
copy numbers is independent of X. To investigate how fluctuations in Y
propagate downstream, a protein Z whose production rate only depend on Y is
introduced. Intriguingly, results show that the extent of noise in Z increases
with noise in X, in spite of the fact that the magnitude of noise in Y is
invariant of X. Such counter intuitive results arise because X enhances the
time-scale of fluctuations in Y, which amplifies fluctuations in downstream
processes. In summary, while feedforward systems can buffer a protein from
noise in its upstream regulators, noise can propagate downstream due to changes
in the time-scale of fluctuations.
