  In living cells, proteins self-assemble into large functional structures
based on specific interactions between molecularly complex patches. Due to this
complexity, protein self-assembly results from a competition between a large
number of distinct interaction energies, of the order of one per pair of
patches. Current self-assembly models however typically ignore this aspect, and
the principles by which it determines the large-scale structure of protein
assemblies are largely unknown. Here, we use Monte-Carlo simulations and
machine learning to start to unravel these principles. We observe that despite
widespread geometrical frustration, aggregates of particles with complex
interactions fall within only a few categories that often display high degrees
of spatial order, including crystals, fibers, and oligomers. We then
successfully identify the most relevant aspect of the interaction complexity in
predicting these outcomes, namely the particles' ability to form periodic
structures. Our results provide a first extensive characterization of the rich
design space associated with identical particles with complex interactions, and
could inspire engineered self-assembling nanoobjects as well as help understand
the emergence of robust functional protein structures.
