  Cells estimate concentrations of chemical ligands in their environment using
a limited set of receptors. Recent work has shown that the temporal sequence of
binding and unbinding events on just a single receptor can be used to estimate
the concentrations of multiple ligands. Here, for a network of many ligands and
many receptors, we show that such temporal sequences can be used to estimate
the concentration of a few times as many ligand species as there are receptors.
Crucially, we show that the spectrum of the inverse covariance matrix of these
estimates has several universal properties, which we trace to properties of
Vandermonde matrices. We argue that this can be used by cells in realistic
biochemical decoding networks.
