Novel matrix descriptor for determination of the connectivity of secondary structure segments in proteins. Analysis of general properties using graph theory Original Research Article

Connectivity of secondary structure segments in proteins is described by the matrix descriptor [sij]. Physical background for design of this super-secondary structure descriptor is summarized, rules for its generation from reference X-ray structures are defined and formal variants of its form are discussed. It is shown that the practical form of [sij] that enumerates the secondary structure segments and their interconnections can be converted into Eulerian graph G. Mathematical analysis of G provides a series of boundary conditions for the descriptor elements which are needed as reliability conditions in practical applications. The number of different sequences of protein segments compatible with a given [sij] is calculated and algorithms for their generation are designed. Finally, a statistical analysis of descriptor elements derived for a reference set of non-homologous protein X-ray structures is used to discover interdependencies between the numbers of respective structural segments (i,j) in native protein folds. It is shown that these interdependencies can be described by a probabilistic logarithmic function.