Very-low-energy complexes generated by random docking of monomeric proteins are predominantly highly symmetrical. (Left) The most symmetrical binding mode of the three lowest-energy structures after high-resolution docking of the normally monomeric proteins 2chy (A) and 2lfo (C). (Right) Low-energy symmetric binding modes after sequence evolution (see SI Appendix): 2chy (B) and 2lfo (D). Sequence optimization lowers the interaction energies and can change the relative energy of different binding modes, but does not fundamentally change their symmetry.

Emergence of symmetry from ensembles of randomly docked homodimeric complexes. (A) Comparison of S_dev distributions for interacting spheres, complexes formed by randomly docking monomeric proteins, and native protein homodimers. Solid blue, numerical solution of supplementary Eq. 1 for contacting spheres; dashed blue, analytic solution (Eq. 1) for spheres at infinite distance; Yellow (1hz6), green (1hz6) and brown (2chy), numerical results for random protein homodimer complexes generated by explicit protein docking. Red, S_dev distribution of 796 naturally occurring protein homodimer structures. The y axis is broken to accommodate the sharp peak near S_dev = 0 for the naturally occurring complexes. (B) Increase in symmetry in random homodimeric complexes with increasingly stringent energy-based selection obtained by numerically integrating Eq. 3. Blue, no energy threshold; black, energy E < −4.6 σ; red, E < − 10.0σ (A more negative threshold implies a tighter binding is required for function.) (C) Numerical results from Rosetta all-atom protein docking calculations with 2chy. Blue, S_dev distribution for randomly generated docked complexes (no energy threshold); black, S_dev distribution for very-low-energy docked complexes with E < −4.6σ below the mean. (D) Increase in the fraction of symmetric structures (S_dev < 0.2 Å) in population after energy-based selection; brown, result from 2chy docking simulations; red, simple model result (SI Appendix, Eq. S8). As discussed in the SI Appendix, we estimate σ to be typically ≈1 kcal/mol (Table S1).