  We typically think of cells as responding to external signals independently
by regulating their gene expression levels, yet they often locally exchange
information and coordinate. Can such spatial coupling be of benefit for
conveying signals subject to gene regulatory noise? Here we extend our
information-theoretic framework for gene regulation to spatially extended
systems. As an example, we consider a lattice of nuclei responding to a
concentration field of a transcriptional regulator (the "input") by expressing
a single diffusible target gene. When input concentrations are low, diffusive
coupling markedly improves information transmission; optimal gene activation
functions also systematically change. A qualitatively new regulatory strategy
emerges where individual cells respond to the input in a nearly step-like
fashion that is subsequently averaged out by strong diffusion. While motivated
by early patterning events in the Drosophila embryo, our framework is
generically applicable to spatially coupled stochastic gene expression models.
