  Random heteropolymers do not display the typical equilibrium properties of
globular proteins, but are the starting point to understand the physics of
proteins and, in particular, to describe their non-native states. So far, they
have been studied only with mean-field models in the thermodynamic limit, or
with computer simulations of very small chains on lattice. After describing a
self-adjusting parallel-tempering technique to sample efficiently the
low-energy states of frustrated systems without the need of tuning the
system-dependent parameters of the algorithm, we apply it to random
heteropolymers moving in continuous space. We show that if the mean interaction
between monomers is negative, the usual description through the random energy
model is nearly correct, provided that it is extended to account for
non-compact conformations. If the mean interaction is positive, such a simple
description breaks out and the system behaves in a way more similar to Ising
spin glasses. The former case is a model for the denatured state of glob- ular
proteins, the latter of naturally-unfolded proteins, whose equilibrium
properties thus result qualitatively different.
