Universal differential equations for quantifying NF-κB-p53 signaling crosstalk

Universal differential equations (UDEs) have emerged as a powerful tool for scientific discovery, uniting differential equations and deep learning. Yet, their application to the noisy and complex data of systems biology remains limited. Here, we demonstrate that UDEs, paired with symbolic regression, can quantify the crosstalk between NF-κB and p53 signaling pathways based on experimental data exhibiting complex, oscillatory dynamics. We validate the framework on synthetic benchmarks, showing that crosstalk recovery is robust to measurement noise and improves markedly with the number of available time series. We then apply the framework to 106 simultaneously measured single-cell p53 and NF-κB time series following DNA damage and NF-κB activation, constituting the largest-scale UDE application in systems biology so far. UDEs with both a detailed and a minimal mechanistic p53 model consistently identify a monotonically increasing crosstalk function in which elevated NF-κB levels enhance p53 synthesis. Symbolic regression distills the learned neural network output into a compact, interpretable closed-form expression, providing a quantitative, time-resolved characterization of NF-κB-driven amplification of p53 at the single-cell level. Our work thus delivers a dual contribution: methodologically, it establishes UDEs as a viable tool for gaining quantitative insights in complex, noisy biological systems at scale; biologically, it provides a data-driven characterization of NF-κB-p53 pathway crosstalk from single-cell dynamics.

Source code is available at: https://github.com/DILiS-lab/ude-crosstalk-discovery.