Pichia strains such as SMD1165, SMD1168 can grow in YPD media or even in synthetic complete yeast media that do not contain any biotin. but BMGY specifically include biotin in its composition. I am trying to figure out what advantage does it provide to the media for cell growth. May be I should just go ahead and prepare BMGY with and without biotin and try expressing the clones I have to see if there is any difference.
I think actually there is enough biotin coming from yeast extract and peptone in YEPD to ensure normal Pichia growth. Synthetic complete also contains biotin coming from the YNB. I think the BMGY recipe is directly inspired from the BMG recipe (which contains biotin) only supplemented with yeast extract and peptone. As you said, it might be that biotin is not necessary in BMGY context (just as in YEPD context).
One of the functions of biotin is that it is the prosthetic group of pyruvate carboxylase, which is essential for gluconeogenesis. Glucose production is necessary (e.g. for cell wall synthesis) when methanol or glycerol is the sole carbon source. Thus the inclusion of more than normal biotin likely reflects a precaution not to limit the yeasts in this essential process. Media with 0,67% YNB will contain 2 microgram / L biotin, standard YPD (1% yeast extract, 2 % peptone) will roughly contain 10-20 microgram / L (mostly deriving from the yeast extract).