Final answer:
A peripheral membrane protein synthesized in the ER lumen would end up on the outside of the plasma membrane after being processed by the Golgi apparatus and transported via vesicles that maintain the membrane's polarity.
Step-by-step explanation:
If a peripheral membrane protein is synthesized in the lumen of the endoplasmic reticulum (ER), it would eventually end up on the outside of the plasma membrane. This process of protein synthesis and sorting involves the rough ER, where the protein is synthesized, and then the protein travels to the Golgi apparatus for modification. This pathway involves transport vesicles that move the protein from the ER to the cis face of the Golgi apparatus, where it's processed and modified. Once this is complete, further transport vesicles pinch off from the trans face of the Golgi and carry the protein to the plasma membrane.
When these vesicles fuse with the plasma membrane, the protein, embedded in the vesicle's own membrane, is incorporated into the plasma membrane with its functional domain facing outward, towards the extracellular space. This orientation is due to the vesicles maintaining the polarity of the membranes as they travel through the endomembrane system. Thus, a peripheral protein that is synthesized within the ER will be exposed on the cell surface once it is processed and integrated into the plasma membrane.