You don't ask questions lightly, do you?
How much do you currently know about Double Well Potential?
The proton would remain positively charged throughout.
Edit To Add:
Here is a wiki page on Nuclear Shell Model.
The section "Deforming the Potential" would have your interest, since a deformed shell relates to the potential for two atoms (example, Hydrogen) to have one well deeper than the other.
In such a state, the proton will have a greater probability of moving to the deeper well. This has been suggested to explain Spontaneous DNA mutation. I'll look for a link in a moment and edit it in.
For now, here is an abstract that covers the gory details using two D modeling.
ETA -2 :
Jeez, all I had to do was stick to wikipedia. It mentions it in the Quantum Tunneling page: