Hi Colin.
This is my Humble opinion.
IF you are not killing yourself daily, i.e. running to such an extent that your body cannot cope, AND you are on the Diet....I agree with the major consensus that it is a bad idea.
HOWEVER, if you use the Atkins diet (Fad or not, it works, and not everyone is as lucky as me when then need to lose weight), and you continue running your four miles a day, I cannot envisage, how or why it would affect you, IF your only plan is to try slim down, so that in the long term you fitness improves.
I'm probably only making sense to myself.
Cheers.