How many proteins are associated with an institution founded after 1880 or an institution with type "Private"?
SELECT count(*) FROM institution AS T1 JOIN protein AS T2 ON T1.institution_id  =  T2.institution_id WHERE T1.founded  >  1880 OR T1.type  =  'Private'
proteins associated with an institution refers to JOIN on institution_id; institution founded after 1880 refers to founded > 1880; institution with type 'Private' refers to type = 'Private'; count proteins refers to COUNT(*);