what role do cell membranes play in protein synthesis or gene expression?

Wow big question!

The nucleus is contained by two membranes (an inner and an outer).  These act as a barrier to keep DNA inside the nucleus where transcription occurs.  The RNA generated during translation is transported out through nuclear pores.


Many proteins are synthesized at the Endoplasmic reticulum.  The ribosomes bind to the ER (a membrane bound organelle) and translate the RNA message into protein that either traverses the membrane (like a protein channel does) or is inside the ER in a vesicle (for soluble proteins that are secreted by the cell by exocytosis... like growth factors and hormone proteins).

The proteins are modified post-translationally in both the ER and the golgi appartus.  They traffic to different places in the cell inside or on vesicles.  For instance proteins move from the ER to the golgi in/on a vesicle (which of course is made up of membrane).