||Proteins in nature exhibit special properties including high regularity in structure and high correlation between the motifs of the amino acid sequences and the corresponding substructures (secondary structures, active sites), which are absent in random sequences. Our study considers a model protein with twenty-seven residues that fold in a lattice space and attain different compact cubic conformations. Every residue of the protein sequence is decorated by two types of model amino acid that represent the twenty amino acids found in nature. A subset of protein sequences was defined for each cubic conformation such that every sequence in the set takes the conformation as its unique ground state. The substructures (the bonding between residues) and the patterns (decoration of amino acid on the residues) on the surfaces of the model protein were analyzed and the existence of preferential patterns on different substructures was observed, which is in agreement with our understanding of real proteins. Our finding of preferential patterns on the substructures demonstrates the strong correlation between protein structures and their sequences of amino acids, and provides a statistical justification for the use of various algorithms for protein structure and protein function prediction from its own sequence.