The board shall consist of eleven persons, seven of whom shall be physicians or surgeons licensed under the laws of the State, two of whom shall be osteopathic physicians licensed under the laws of the State, and two of whom shall be lay members appointed from the public at large. Of the nine members who are physicians, surgeons, or osteopathic physicians, at least five shall be appointed from the city and county of Honolulu and at least one shall be appointed from each of the other counties. Medical societies in the various counties may conduct elections periodically but no less frequently than every two years to determine nominees for the board to be submitted to the governor. In making appointments, the governor may consider recommendations submitted by the medical societies and the public at large. Each member shall serve until a successor is appointed and qualified.