gene expression being turned off, or on is most certainly about diet... polyphenols are found to have an anti-cancer effect, stimulating selective apoptotic cell death in cells expressing HER-2.
It is a mistake to get hung up on any one thing for prevention...
In other words, you cannot depend solely on Iodine and disregard your diet, co-factors that enable Iodine to be more efficacious, e.g. other minerals, environment, and lifestyle...