New from old?

Take a portion of your white starter and start feeding it an increasing percentage of whole wheat and a decreasing percentage of white flour.  For instance, do a couple of feedings with 20% whole wheat and 80% white.  Then a couple of feedings at 40% whole wheat and 60% white.  And so on until you are feeding 100% whole wheat.  That will give the yeasts and bacteria in your starter an opportunity to adapt to the changing conditions instead of hitting them with an entirely different food source all at once.

That said, sourdough starters are remarkably resilient.  If you did go to feeding it straight whole wheat, it might be somewhat sluggish for a few days but would eventually bounce back to full strength.

Do expect that the whole wheat starter will smell and taste different than it's white brother.  The whole wheat brings a lot more and different compounds to the party than does the white flour.  Once it stabilizes, the population of microbial life that thrives on the whole wheat will also be different than the population that thrives in the white starter.  You may also notice that the whole wheat starter ripens faster and requires more-frequent feedings as a result.  Or not.

Paul