Institute in Split developing part of flower (6)
I believe the answer is:
pistil
'part of flower' is the definition.
(I know this)
'institute in split developing' is the wordplay.
'institute' becomes 'i' (common abbreviation as in MIT).
'in' means one lot of letters goes inside another.
'developing' indicates an anagram (letters developed into a new form).
'split' is an anagram of 'pistl'.
'i' going inside 'pistl' is 'PISTIL'.
(Other definitions for pistil that I've seen before include "Female part of a plant" , "Female organ of a flower" , "Stigma, style and ovary" , "Seed-producing part of a flower" , "Female organ in plant" .)