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.
Highlights
MotivationSupramolecular assembly is prevalent in nature, healthcare and engineering, but poorly understood
The stability and binding affinity of subassemblies depend on free energy, whose landscape in the case of assembly is heavily influenced by configurational entropy; this depends on accurate computation of configurational volumes by sampling, attempted by a long and distinguished series of methods [5,6,7,8,9,10,11,12,13]
This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres under short-range potentials. It is motivated by a longer term goal to exploit natural symmetries using assembly trees and other concepts described in the previous sections that have appeared in various avatars in the community, including our work on efficient atlasing and search of assembly landscapes (EASAL). Such an understanding of symmetries is essential for significantly reducing the complexity of the computation of configurational and combinatorial entropy, as well as kinetics, since spheres within rigid bunches of an assembly system could be identical and bunches could be identical, as well, giving underlying symmetry groups of large order, which grow with the number of participating spheres and bunches
Summary
Supramolecular assembly is prevalent in nature, healthcare and engineering, but poorly understood. A non-pairwise component of the potential energy function is in the form of global potential energy terms that capture the tethers between the rigid components within a monomer, as well as other global potential energy terms that implicitly represent the solvent (water or lipid bilayer membrane) effect [2,3,4] These are independent of particular pairs of atoms. Output: prediction of those interactions that are crucial for the assembly process to terminate in the given input assembly configuration. Output: prediction of minimal alterations of the building blocks or interactions that would significantly increase the likelihood of the assembly process terminating in the given input assembly configuration. Computer simulations guided by theoretical first principles and standard paradigms, such as Monte Carlo (MC) or molecular dynamics (MD), are limited due to the reasons detailed in the subsections
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