  Proteins populate a manifold in the high-dimensional sequence space whose
geometrical structure guides their natural evolution. Leveraging
recently-developed structure prediction tools based on transformer models, we
first examine the protein sequence landscape as defined by the folding score
function. This landscape shares characteristics with optimization challenges
encountered in machine learning and constraint satisfaction problems. Our
analysis reveals that natural proteins predominantly reside in wide, flat
minima within this energy landscape. To investigate further, we employ
statistical mechanics algorithms specifically designed to explore regions with
high local entropy in relatively flat landscapes. Our findings indicate that
these specialized algorithms can identify valleys with higher entropy compared
to those found using traditional methods such as Monte Carlo Markov Chains. In
a proof-of-concept case, we find that these highly entropic minima exhibit
significant similarities to natural sequences, especially in critical key sites
and local entropy. Additionally, evaluations through Molecular Dynamics
suggests that the stability of these sequences closely resembles that of
natural proteins. Our tool combines advancements in machine learning and
statistical physics, providing new insights into the exploration of sequence
landscapes where wide, flat minima coexist alongside a majority of narrower
minima.
