  Signal propagation in biochemical networks is characterized by the inherent
randomness in gene expression and fluctuations of the environmental components,
commonly known as intrinsic and extrinsic noise, respectively. We present a
theoretical framework for noise propagation in a generic two-step cascade
(S$\rightarrow$X$\rightarrow$Y) regarding intrinsic and extrinsic noise. We
identify different channels of noise transmission that regulate the individual
and the overall noise properties of each component. Our analysis shows that the
intrinsic noise of S alleviates the general noise and information transmission
capacity along the cascade. On the other hand, the intrinsic noise of X and Y
acts as a bottleneck of information transmission. We also show a hierarchical
relationship among the intrinsic noise levels of S, X, and Y, with S exhibiting
the highest level of intrinsic noise, followed by X and then Y. This hierarchy
is preserved within the two-step cascade, facilitating the highest information
transmission from S to Y via X.
