Step 1: Understanding the Question:
The question asks to pinpoint the exact intracellular compartment where $\beta$-oxidation of fatty acids takes place in eukaryotic cells.
Step 2: Key Formula or Approach:
Fatty acid breakdown proceeds through four repeating biochemical steps (oxidation, hydration, oxidation, thiolysis) that convert fatty acyl-CoA into acetyl-CoA units inside the mitochondrion.
Step 3: Detailed Explation:
• Fatty Acid Activation: Fatty acids are first activated in the cytosol to form fatty acyl-CoA by acyl-CoA synthetase.
• Carnitine Shuttle System: Long-chain fatty acyl-CoA molecules are transported across the inner mitochondrial membrane via the carnitine palmitoyltransferase (CPT-I CPT-II) shuttle mechanism.
• Localization of $\beta$-Oxidation Enzymes: Once inside the mitochondrial matrix, soluble enzymes (Acyl-CoA dehydrogese, Enoyl-CoA hydratase, 3-hydroxyacyl-CoA dehydrogese, and $\beta$-ketoacyl-CoA thiolase) degrade the fatty acyl chain two carbons at a time.
• Yield: Each cycle produces 1 molecule of Acetyl-CoA (which enters the TCA cycle located in the matrix), 1 molecule of DH, and 1 molecule of FADH$_2$.
Step 4: Fil Answer:
The enzymes responsible for $\beta$-oxidation of fatty acids reside within the mitochondrial matrix.