A topology for the space of protein chains and a notion of local statistical stability for their three-dimensional structures

The present paper aims at introducing and discussing to some extent the possibility of endowing the set of all possible protein chains–which, being obviously empirically unknown, will be thought of as a mathematical object, i.e.   a sample space ΩPΩP–with the structure of a topological space. The main goal of the discussion is to try to identify some kind of (statistical) law linking classes of sequences to distributions of suitably defined local geometric properties of the chains folded in the native configurations of proteins. This should enable one to translate the physical multiple interactions leading to the final configuration of a protein into stochastically defined multiple interactions between individuals whose states are described by geometric parameters only. To achieve our goal, we need a suitable notion of (pseudo-)distance between primary structures, which could improve the one introduced in a previous paper, and allow us to give a corresponding definition of stability of local geometric structures (and, as a consequence, of three-dimensional configurations) of proteins.