This question is actually extremely deep. I think it's all about connections. The actual atoms may change, but the connections remain for the most part in tact. These connections of which I speak are the physical bones, muscles, etc., and also the synapses in our brain.
The question arises, does a new born baby have a "self"? I highly doubt that any such child has a concept of identity or of the self, and yet they do respond to stimuli such as pain and heat, etc. That same child is now you.
What we call genetics, nature, is connections of DNA. What we call experience, nurture, is connections of synapses in the brain (for the most part). These connections can be built upon and modified to a certain extent, but they all unravel upon death.
The extent to which these connection remain viable during life determine our idea of the "self".
(My opinion in a nutshell)