  We develop numerical and analytical approaches to calculate mutual
information between complete paths of two molecular components embedded into a
larger reaction network. In particular, we focus on a continuous-time Markov
chain formalism, frequently used to describe intracellular processes involving
lowly abundant molecular species. Previously, we have shown how the path mutual
information can be calculated for such systems when two molecular components
interact directly with one another with no intermediate molecular components
being present. In this work, we generalize this approach to biochemical
networks involving an arbitrary number of molecular components. We present an
efficient Monte Carlo method as well as an analytical approximation to
calculate the path mutual information and show how it can be decomposed into a
pair of transfer entropies that capture the causal flow of information between
two network components. We apply our methodology to study information transfer
in a simple three-node feedforward network, as well as a more complex positive
feedback system that switches stochastically between two metastable modes.
