  Emergence of new protein structures has proved difficult to trace in nature
and engineer in the laboratory. However, one aspect of structure evolution has
proved immensely helpful for determining the three-dimensional structure of
proteins from their sequences: in the vast majority of cases, proteins that
share more than 30% sequence identity have similar structures. Below this mark
is the "twilight zone" where proteins may have identical or very different
structures. These observations form the foundational intuition behind structure
homology modeling. Despite their importance, however, they have never received
a comprehensive biophysical justification. Here we show that the onset of the
twilight zone is more gradual for proteins with low contact density, a proxy
for low thermodynamic stability, than proteins with high contact density. Then
we present an analytical model that treats divergent fold evolution as an
activated process, in analogy to chemical kinetics, where sequence evolution
must overcome thermodynamically unstable evolutionary intermediates to discover
new folds. This model explains the existence of a twilight zone and explains
why its onset is more abrupt for some classes of proteins than for others. We
test the assumptions of the model and characterize the dynamics of fold
evolution using evolutionary simulations of model proteins and cell
populations. Overall these results show how fundamental biophysical constraints
directed evolutionary dynamics leading to the Universe of modern protein
structures and sequences.
