NF-$\kappa $B dynamics show digital activation and analog information processing in cells

Cells operate in ever changing environments using extraordinary communication capabilities. Cell-to-cell communication is mediated by signaling molecules that form spatiotemporal concentration gradients, which requires cells to respond to a wide range of signal intensities. We used high-throughput microfluidic cell culture, quantitative gene expression analysis and mathematical modeling to investigate how single mammalian cells respond to different concentrations of the signaling molecule TNF-$\alpha$ via the transcription factor NF-$\kappa $B. We measured NF-$\kappa$B activity in thousands of live cells under TNF-$\alpha $ doses covering four orders of magnitude. In contrast to population studies, the activation is a stochastic, switch-like process at the single cell level with fewer cells responding at lower doses. The activated cells respond fully and express early genes independent of the TNF-$\alpha $ concentration, while only high dose stimulation results in the expression of late genes. Cells also encode a set of analog parameters such as the NF-$\kappa $B peak intensity, response time and number of oscillations to modulate the outcome. We developed a stochastic model that reproduces both the digital and analog dynamics as well as the gene expression profiles at all measured conditions, constituting a broadly applicable model for TNF-$\alpha $ induced NF-$\kappa$B signaling in various types of cells.