On representation of proteins by star-like graphs

To arrive at graphical representations of proteins one is confronted with number of arbitrary decisions how to assign the 20 natural amino acids to equivalent or non-equivalent sites of underlying geometrical objects used for construction of their graphical representation. Here we consider representation of proteins based on generalized star graphs, which are graphs with one vertex of maximal degree in the center to which are attached other vertices of either degree one or two. The matrix representation of proteins based on star-like graphs has an important advantage in that, while its pictorial representation depends on selected assignment of amino acids to various branches of star graph, its properties do not depend on the adopted assignment of vertices to amino acids. Hence, the derived graph invariants, devoid of artifacts associated with graphical representations of biosequences, will better reflect upon the inherent properties of protein structure. We describe several graph invariants, mostly extracted from distance matrices of star-like graphs, which can serve as protein descriptors. The approach is illustrated on strand A of the human insulin.