Koch-Tataru solutions are global-in-time small data mild solutions to the 3D NSE emanating from small data in $BMO^{-1}$. There have been several recent works in which the regularizing-decay rate estimates for Koch-Tataru solutions have been obtained. Spatial analyticity of solutions then follows as a consequence. I will present a different approach in which the spatial analyticity is obtained directly together with explicit estimates on the time-evolution of the domain of analyticity. The regularizing-decay rate estimates will then follow at once.