Symmetry in Sphere-based Assembly Configuration Spaces

Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates correspond to various types of macrostates can significantly reduce the complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres with the following goals. (1) We give new, formal definitions of various concepts relevant to sphere-based assembly that occur in previous work, and in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously developed concepts include, for example, (a) assembly configuration spaces, (b) stratification of assembly configuration space into regions defined by active constraint graphs, (c) paths through the configurational regions, and (d) coarse assembly pathways. (2) We demonstrate the new symmetry concepts to compute sizes and numbers of orbits in two example settings appearing in previous work. (3) We give formal statements of a variety of open problems and challenges using the new conceptual definitions.