Before revision:  By Monte Carlo simulations of a statistical mechanical model of protein sequence alignment that coherently incorporates both short-range and long-range interactions in addition to variable-length insertions  , we demonstrate the existence of such transition between natural-like sequences and random sequences in the sequence subspaces for 15 domain families  .
After revision:  By Monte Carlo simulations of a statistical mechanical model of protein sequence alignment that coherently incorporates both short-range and long-range interactions in addition to variable-length insertions to reproduce the statistics of the multiple sequence alignment of a given protein family , we demonstrate the existence of such transition between natural-like sequences and random sequences in the sequence subspaces for 15 domain families .