  Signaling in enzymatic networks is typically triggered by environmental
fluctuations, resulting in a series of stochastic chemical reactions, leading
to corruption of the signal by noise. For example, information flow is
initiated by binding of extracellular ligands to receptors, which is
transmitted through a {cascade involving} kinase-phosphatase stochastic
chemical reactions. For a class of such networks, we develop a general
field-theoretic approach in order to calculate the error in signal transmission
as a function of an appropriate control variable. Application of the theory to
a simple push-pull network, a module in the kinase-phosphatase cascade,
recovers the exact results for error in signal transmission previously obtained
using umbral calculus (Phys. Rev. X., {\bf 4}, 041017 (2014)). We illustrate
the generality of the theory by studying the minimal errors in noise reduction
in a reaction cascade with two connected push-pull modules. Such a cascade
behaves as an effective {three-species} network with a pseudo intermediate. In
this case, optimal information transfer, resulting in the smallest square of
the error between the input and output, occurs with a time delay, which is
given by the inverse of the decay rate of the pseudo intermediate. There are
substantial deviations between simulations and predictions of the linear theory
in error in signal propagation in an enzymatic push-pull network for a certain
range of parameters. Inclusion of second order perturbative corrections shows
that differences between simulations and theoretical predictions are minimized.
Our study establishes that {a} field theoretic formulation {of} stochastic
biological {signaling offers} a systematic way to understand error propagation
in networks of arbitrary complexity.
