  Repeat proteins are made with tandem copies of similar amino acid stretches
that fold into elongated architectures. Due to their symmetry, these proteins
constitute excellent model systems to investigate how evolution relates to
structure, folding and function. Here, we propose a scheme to map evolutionary
information at the sequence level to a coarse-grained model for repeat-protein
folding and use it to investigate the folding of thousands of repeat-proteins.
We model the energetics by a combination of an inverse Potts model scheme with
an explicit mechanistic model of duplications and deletions of repeats to
calculate the evolutionary parameters of the system at single residue level.
This is used to inform an Ising-like model that allows for the generation of
folding curves, apparent domain emergence and occupation of intermediate states
that are highly compatible with experimental data in specific case studies. We
analyzed the folding of thousands of natural Ankyrin-repeat proteins and found
that a multiplicity of folding mechanisms are possible. Fully cooperative
all-or-none transition are obtained for arrays with enough sequence-similar
elements and strong interactions between them, while non-cooperative
element-by-element intermittent folding arose if the elements are dissimilar
and the interactions between them are energetically weak. In between, we
characterised nucleation-propagation and multi-domain folding mechanisms.
Finally, we showed that stability and cooperativity of a repeat-array can be
quantitatively predicted from a simple energy score, paving the way for guiding
protein folding design with a co-evolutionary model.
