Atlasing of Assembly Landscapes using Distance Geometry and Graph   Rigidity
  We describe a novel geometric methodology for analyzing free-energy and kinetics of assembly driven by short-range pair-potentials in an implicit solvent, and provides illustrations of its unique capabilities. An atlas is a labeled partition of the assembly landscape into a topological roadmap of maximal, contiguous, nearly-equipotential-energy conformational regions or macrostates, together with their neighborhood relationships. The new methodology decouples the roadmap generation from sampling and produces: (1) a query-able atlas of local potential energy minima, their basin structure, energy barriers, and neighboring basins; (2) paths between a specified pair of basins; and (3) approximations of relative path lengths, basin volumes (configurational entropy), and path probabilities. Results demonstrating the core algorithm's capabilities have been generated by a resource-light, opensource software implementation EASAL. EASAL atlases several hundred thousand macrostates in minutes on a standard laptop. Subsequent path and basin computations each take seconds. The core algorithm's correctness, time complexity, and efficiency-accuracy tradeoffs are formally guaranteed using modern geometric constraint systems. The methodology further links geometric variables of the input assembling units to a type of intuitive topological bar-code of the output atlas, which in turn determine stable assembled structures and kinetics. This succinct input-output relationship facilitates reverse analysis, and control towards design. We use the novel convex Cayley (distance-based) parametrization that is unique to assembly, as opposed to folding. Sampling microstates with macrostate-specific Cayley parameters avoids gradient-descent search used by all prevailing methods. This increases sampling efficiency, significantly reduces the number of repeated and discarded samples. 
Rahul Prabhu and Meera Sitharam and Aysegul Ozkan and Ruijin Wu
cs.CG