**Core Concept**
Sorting of protein molecules is a critical process in cellular biology, referring to the mechanism by which cells direct proteins to specific destinations within the cell or outside of it. This process is essential for maintaining cellular homeostasis and function. The sorting process involves the recognition of specific signals on the protein molecules by the cell's machinery.

**Why the Correct Answer is Right**
The sorting of protein molecules occurs in the _trans-Golgi network_ (TGN), a region of the Golgi apparatus where proteins are sorted and packaged for secretion or targeted to specific cellular compartments. The TGN contains a complex network of tubules and vesicles that facilitate the sorting and packaging process. Proteins are recognized by specific receptors and adaptors that bind to signals on the protein surface, directing them to the correct destination.

**Why Each Wrong Option is Incorrect**
* **Option A:** The _endoplasmic reticulum_ (ER) is involved in protein synthesis and folding, but it is not the primary site for protein sorting.
* **Option B:** The _mitochondria_ are involved in energy production and have their own protein synthesis machinery, but they are not involved in protein sorting.
* **Option C:** The _nucleus_ is involved in DNA replication and transcription, but it is not directly involved in protein sorting.

**Clinical Pearl / High-Yield Fact**
The trans-Golgi network is a critical site for the sorting and packaging of proteins, and defects in this process can lead to a range of diseases, including lysosomal storage disorders.

**Correct Answer:** C. The Golgi apparatus (specifically the trans-Golgi network) is responsible for sorting and packaging proteins for secretion or targeting to specific cellular compartments.