Learning molecular traits of human pain disease via voltage-gated sodium channel structure renormalization

Mammalian neurophysiology vitally depends on the stable functioning of transmembrane, pore-forming voltage-sensing proteins known as voltage-gated sodium channels (NaVChs). Deciphering the principles of NaVCh spatial organization can illuminate fundamental structure-function aspects of pore-forming proteins and offer new opportunities for pharmacological treatment of associated diseases such as chronic pain. Here, we introduce a renormalization group flow paradigm permitting a formal investigation of NaVCh thermostability properties. Our procedures are solidified by deriving an atom-packing entropy and validated over 121 experimentally resolved NaVCh structures of prokaryotic and eukaryotic origin. We uncover the universality of a critical inflection point regulating the thermostability of the pore domain relative to the voltage sensors, summarized in terms of a generalized Widom scaling law. A machine learning algorithm, rationalized in terms of the violation of inertia and conductivity channel constraints, identifies pain-disease-associated mutation hotspots in the human NaV1.7 channel. Our work illustrates how first-principles-based machine learning approaches can deliver accurate insights for human pain medicine and clinicians at a reduced computational cost, while clarifying the self-organized critical nature of NaVChs.