Silk is one of the most intriguing examples of biomolecular self-assembly, yet little is understood of molecular mechanisms behind the flow behavior generating these complex high-performance fibers. This work applies the polymer physics of entangled solution rheology to present a first microphysical understanding of silk in the linear viscoelastic regime. We show that silk solutions can be approximated as reptating polymers with "sticky" calcium bridges whose strength can be controlled through the potassium concentration. This approach provides a new window into critical microstructural parameters, in particular identifying the mechanism by which potassium and calcium ions are recruited as a powerful viscosity control in silk. Our model constitutes a viable starting point to understand not only the "flow-induced self-assembly" of silk fibers but also a broader range of phenomena in the emergent field of material-focused synthetic biology.