Capacity estimates of additive inverse Gaussian molecular channels with relay characteristics

Molecular communications is an emergent field that seeks to develop nanoscale communication devices using design principles gleaned from studies of the topology and dynamic properties of biological signaling networks. To understand how these networks function, we must first characterize the functional building blocks that compose them, and the best candidates for those are the topologically distinct subnetworks, or motifs, that appear in a statistically improbable abundance. In transcriptional networks, one of the most prevalent motifs is the feed-forward loop, a three node motif wherein one top-level protein regulates the expression of a target gene either directly or indirectly through an intermediate regulator protein. Currently, no systematic effort has been made to treat an isolated feed-forward loop as a stand-alone signal amplifying/attenuating device and understand its communication capacity in terms of the diffusion of individual molecules. To address this issue, we derive a theorem that estimates the upper and lower bounds of the channel capacity for a relay channel, which structurally corresponds to a feed-forward loop, by using an additive inverse Gaussian noise channel model of protein-ligand binding. Our results are just a first step towards assessing the performance bounds of simplified biological circuits in order to guide the development and optimization of synthetic, bio-inspired devices that can be used as information processing and forwarding units.