# Universal properties of concentration sensing in large ligand-receptor networks

## TL;DR

This paper demonstrates that large ligand-receptor networks can estimate multiple ligand concentrations from temporal binding sequences, revealing universal properties linked to Vandermonde matrices that can inform cellular biochemical decoding.

## Contribution

It introduces a theoretical framework showing how cells can decode multiple ligand concentrations using receptor networks, highlighting universal spectral properties of the inverse covariance matrix.

## Key findings

- Temporal sequences enable estimation of multiple ligands.
- Universal spectral properties linked to Vandermonde matrices.
- Potential implications for cellular biochemical decoding.

## Abstract

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.