Synthetic protein nanocages are engineered macromolecular structures designed to mimic natural capsids for targeted drug delivery and metal-ion sequestration. A newly developed nanocage consists of multiple protein subunits assembling into a hollow spherical shell to isolate toxic zinc ions (Zn2+\text{Zn}^{2+}Zn2+) from the cytoplasm.
A single engineered protein nanocage has a hollow inner cavity with a total volume of 320 nm3320\text{ nm}^3320 nm3.
This inner cavity can encapsulate a maximum of 250025002500 Zn2+\text{Zn}^{2+}Zn2+ ions.
The effective volume of a single Zn2+\text{Zn}^{2+}Zn2+ ion is 8.60×10−3 nm38.60 \times 10^{-3}\text{ nm}^38.60×10−3 nm3.
Calculate the volume of the inner cavity that is not occupied by Zn2+\text{Zn}^{2+}Zn2+ ions when the nanocage is at maximum capacity.
Explain how the hydrophobic and hydrophilic interactions of the amino acid R-groups stabilize the folded, hollow shell structure of this protein nanocage in an aqueous environment.
