The past decade has seen great advances in our understanding of the role of
noise in gene regulation and the physical limits to signaling in biological
networks. Here we introduce the spectral method for computation of the joint
probability distribution over all species in a biological network. The spectral
method exploits the natural eigenfunctions of the master equation of
birth-death processes to solve for the joint distribution of modules within the
network, which then inform each other and facilitate calculation of the entire
joint distribution. We illustrate the method on a ubiquitous case in nature:
linear regulatory cascades. The efficiency of the method makes possible
numerical optimization of the input and regulatory parameters, revealing design
properties of, e.g., the most informative cascades. We find, for threshold
regulation, that a cascade of strong regulations converts a unimodal input to a
bimodal output, that multimodal inputs are no more informative than bimodal
inputs, and that a chain of up-regulations outperforms a chain of
down-regulations. We anticipate that this numerical approach may be useful for
modeling noise in a variety of small network topologies in biology