Density Physics-Informed Neural Network reveals sources of cell heterogeneity in signal transduction

The transduction time between signal initiation and final response provides valuable information on the underlying signaling pathway, including its speed and precision. Furthermore, multimodality in transduction-time distribution informs that the response is regulated by multiple pathways with different transduction speeds. Here, we developed Density physics-informed neural network (Density-PINN) to infer the transduction-time distribution, challenging to measure, from measurable final stress response time traces. We applied Density-PINN to single-cell gene expression data from 16 promoters regulated by unknown pathways in response to antibiotic stresses. We found that promoters with slower signaling initiation and transduction exhibit larger cell-to-cell heterogeneity in response intensity. However, this heterogeneity was greatly reduced when the response was regulated by slow and fast pathways together. This suggests a strategy for identifying effective signaling pathways for consistent cellular responses to disease treatments. Density-PINN can also be applied to understand various time delay systems, including infectious diseases.