A combinatorial approach to the problem of self-assembly

A formal scheme is proposed in order to mathematically describe some real features of the biological assembly. The assembly is considered to be a process of pairwise interactions of subunits leading to the creation of structures. If only one structure can be assembled according to the given interaction rules, it is called unique. Unique structures are interpreted as formal analogs of self-assembling biological structures. A criterion is found which can be used to develop an effective algorithm for recognizing the uniqueness of finite structures. Symmetry of finite and infinite unique structures is investigated. The relation is determined between the symmetry group of an infinite unique structure and its packing in space which enables classification of such structures in crystallographic terms. The applications of the formal scheme are discussed.