We study the protein structure problem in a prescribed, non-compact volume (with a hole). The choice of such a volume offers the advantage of being closer in structure to real proteins, Using an exact enumeration technique we investigate the thermodynamic properties of the chain’s configurations, These findings are compared to the results obtained for fully compact structures,such as the 3 x 3 x 3 cube. Interestingly we find here deviations from the random energy model.