Drake became the fifth chancellor of the University of California, Irvine in July 2005. Previously, he served as UC Vice President for Health Affairs, overseeing education and research activities at UC's 15 health sciences schools. In that capacity, he oversaw the UC Special Research Programs, in tobacco-related disease, breast cancer and HIV/AIDS research; and the California/Mexico Health Initiative and the California Health Benefits Review Program. Prior to his appointment in the Office of Health Affairs, he was the Steven P. Shearing Professor of Ophthalmology and senior associate dean at the UC San Francisco School of Medicine.

Drake has received numerous awards for teaching, public service, mentoring and research, including the UCSF School of Medicine's Clinical Teaching Award, and the UCSF Chancellor's Award for Public Service. He was the UCSF alumnus of the year in 2000, and was inducted into the Gold Headed Cane Society in 2003. He is the recipient of the Herbert W. Nickens Award from the American Association of Medical Colleges, in recognition of his career-long efforts to promote "social justice through medical education."

He is an active member of several national scientific and scholarly societies including the Institute of Medicine. His is the immediate past national president of the Alpha Omega Alpha Honor Medical Society, and is chair-elect of the board of trustees of the Association of Academic Health Centers. He has written dozens of scholarly articles and chapters, and has co-authored four books.