Symmetry in Sphere-Based Assembly Configuration Spaces

Abstract

Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring 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 that correspond to various types of macrostates can significantly increase efficiency and accuracy, i.e., reduce the notorious 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, as well as accuracy vs. efficiency trade-offs 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 the sphere-based assembly setting 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: (i) assembly configuration spaces; (ii) stratification of assembly configuration space into configurational regions defined by active constraint graphs; (iii) paths through the configurational regions; and (iv) coarse assembly pathways. (2) We then demonstrate the new symmetry concepts to compute the sizes and numbers of orbits in two example settings appearing in previous work. (3) Finally, we give formal statements of a variety of open problems and challenges using the new conceptual definitions.Keywords: sphere assembly; configuration space; stratification; distance constraints; Cayley geometry; entropy; kinetics; pathways MotivationSupramolecular assembly is prevalent in nature, healthcare and engineering, but poorly understood. The assembly starts with identical copies of structures drawn from a small number of types. Modeling these starting structures as rigid bunches of spheres is well suited to assembly processes driven by so-called short-range or hard sphere interaction potentials.More formally, an input to a computational model of an assembly process is an assembly system consisting of the following:

Baseline Comparisons of Complementary Sampling Methods for Assembly Driven by Short-Ranged Pair Potentials toward Fast and Flexible Hybridization

“…It also describes the preliminary extensions of EASAL beyond dimer assemblies and viability of using EASAL for atlasing a wide variety of assembly systems including clusters of up to 24 assembling spherical particles. Leveraging symmetry of the assembly landscape arising from identical input rigid molecular components is discussed in ref 79. The key ideas presented in the above contributions are sketched in Section 2.…”

Efficient Atlasing and Search of Configuration Spaces of Point-Sets Constrained by Distance Intervals

“…This work exploited symmetries and utilized the recursive decomposition of the large viral capsid assembly into an assembly pathway of smaller assembly intermediates. Adapting EASAL to exploit symmetries was the subject of [54].…”

“…4. EASAL screen shots and movies have been used in the papers [54,44,59,53,42,45,60] to illustrate definitions and theoretical results.…”

“…By small assemblies we mean constraint problems with n ≤ 5000 and k = 2; or n ≤ 3 and k ≤ 18. Efficiency can improve significantly when the point-sets are identical, by exploiting symmetries in the configuration space [54].…”

Calculating the Symmetry Number of Flexible Sphere Clusters

“…they don't move far from a reference configurationthe definition coincides with the geometrical symmetries of the configuration, such as a symmetry upon reflection or upon rotation by certain angles around certain axes. Much effort has gone into developing methods to evaluate the geometrical symmetry group of molecules, an effort which is ongoing since this computation is a challenge for large molecules [2,3,4,5,6].…”