Optimal information transfer in enzymatic networks: A field theoretic formulation

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.