The GK does not produce any kind of overdrive or distortion. The Boost control does add some kind of grit or growl. It is hard to explain what it is, but it is obvious when you hear it. But, I would go so far as to call it overdrive, much less distortion. I love the tone of GKs, but the "issues" they have had with the MB2-500 scare me away a little bit. Plus, if you are looking for overdrive from a GK (other than the 2001RB or Backline 300) or, you are likely to be disappointed.
If you want a micro with a some OD/distortion, the ShuttleMAX series from Genz Benz is probably your best bet. They actually design their tube preamp in a way that allows you to push the tube into overdrive without sending a clipped signal to the power section. However, another idea would be to get a Tech 21 VT Bass pedal, which can be used to emulate the tube distortion of an SVT head, and run that in front of your Markbass. Even some die-hard tube snobs (I use this term affectionately

) on here have been surprised at how convincing the tones you can get from that pedal are.